Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623662 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 128 points:
(1,2)(3,17)(4,23)(5,28)(6,33)(7,16)(8,15)(9,13)(10,20)(11,26)(12,31)(14,37)(18,62)(19,68)(21,74)(22,61)(24,46)(25,83)(27,56)(29,90)(30,93)(32,58)(34,57)(35,54)(36,52)(38,55)(39,47)(40,53)(41,50)(42,98)(43,92)(44,73)(45,95)(48,87)(49,71)(51,94)(59,65)(60,70)(63,114)(64,75)(66,112)(67,107)(69,88)(72,103)(76,81)(77,86)(78,122)(79,108)(80,89)(82,105)(84,100)(85,106)(91,102)(96,109)(97,110)(99,126)(101,118)(104,119)(111,116)(113,115)(117,127)(120,123)(121,125)(124,128);(1,3)(2,10)(4,18)(5,29)(6,34)(7,21)(8,40)(9,22)(11,42)(12,49)(13,44)(14,53)(15,45)(16,57)(17,33)(19,30)(20,37)(23,75)(24,78)(25,84)(26,65)(27,67)(28,88)(31,91)(32,92)(35,69)(36,73)(38,95)(39,66)(41,72)(43,51)(46,104)(47,100)(48,107)(50,68)(52,74)(54,102)(55,61)(56,101)(58,60)(59,77)(62,83)(63,79)(64,86)(70,96)(71,94)(76,111)(80,117)(81,126)(82,123)(85,119)(87,98)(89,114)(90,93)(97,118)(99,106)(103,109)(105,113)(108,122)(110,112)(115,124)(116,125)(120,121)(127,128);(1,4,9,27)(2,11,13,47)(3,18,22,67)(5,24,32,82)(6,25,38,87)(7,39,8,26)(10,42,44,100)(12,46,50,105)(14,48,52,83)(15,56,16,23)(17,59,61,112)(19,63,71,116)(20,64,73,118)(21,66,40,65)(28,89,58,76)(29,78,92,123)(30,79,94,125)(31,81,41,80)(33,77,55,110)(34,84,95,98)(35,85,96,124)(36,97,37,86)(43,99,90,127)(45,101,57,75)(49,104,68,113)(51,106,93,128)(53,107,74,62)(54,108,109,121)(60,111,88,114)(69,119,70,115)(72,117,91,126)(102,122,103,120);(1,5)(2,12)(3,19)(4,24)(6,35)(7,31)(8,41)(9,32)(10,43)(11,46)(13,50)(14,54)(15,28)(16,58)(17,60)(18,63)(20,72)(21,69)(22,71)(23,76)(25,85)(26,80)(27,82)(29,34)(30,37)(33,51)(36,94)(38,96)(39,81)(40,70)(42,99)(44,90)(45,102)(47,105)(48,108)(49,53)(52,109)(55,93)(56,89)(57,103)(59,111)(61,88)(62,113)(64,117)(65,115)(66,119)(67,116)(68,74)(73,91)(75,120)(77,106)(78,84)(79,86)(83,121)(87,124)(92,95)(97,125)(98,123)(100,127)(101,122)(104,107)(110,128)(112,114)(118,126);(2,13)(10,44)(11,47)(12,50)(14,52)(15,16)(17,61)(23,56)(28,58)(33,55)(42,100)(43,90)(45,57)(46,105)(48,83)(49,68)(51,93)(53,74)(54,109)(59,112)(60,88)(62,107)(75,101)(76,89)(77,110)(99,127)(102,103)(104,113)(106,128)(108,121)(111,114)(120,122)Polycyclic presentation:
2^9|c^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,g^a=gi,h^a=hi,d^b=df,f^b=fh,h^b=hi,g^c=gi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530860);