Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623668 |
| 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 | |
1882 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)(10,21)(11,26)(12,31)(14,37)(18,62)(19,68)(20,61)(22,74)(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)(43,91)(44,95)(45,72)(48,87)(49,69)(51,94)(59,66)(60,71)(63,114)(64,107)(65,75)(67,112)(70,88)(73,103)(76,81)(77,86)(78,122)(79,108)(80,89)(82,105)(84,101)(85,106)(92,102)(96,109)(97,110)(99,126)(100,117)(104,119)(111,115)(113,116)(118,127)(120,123)(121,125)(124,128);(1,3)(2,10)(4,18)(5,29)(6,34)(7,22)(8,40)(9,20)(11,42)(12,49)(13,45)(14,53)(15,44)(16,57)(17,33)(19,30)(21,37)(23,75)(24,78)(25,84)(26,66)(27,64)(28,88)(31,92)(32,91)(35,70)(36,72)(38,95)(39,67)(41,73)(43,51)(46,104)(47,101)(48,107)(50,68)(52,74)(54,102)(55,61)(56,100)(58,60)(59,77)(62,83)(63,79)(65,86)(69,94)(71,96)(76,111)(80,118)(81,126)(82,123)(85,119)(87,98)(89,114)(90,93)(97,117)(99,106)(103,109)(105,113)(108,122)(110,112)(115,125)(116,124)(120,121)(127,128);(1,4,9,27)(2,11,13,47)(3,18,20,64)(5,24,32,82)(6,25,38,87)(7,39,8,26)(10,42,45,101)(12,46,50,105)(14,48,52,83)(15,56,16,23)(17,59,61,112)(19,63,69,115)(21,65,72,117)(22,67,40,66)(28,89,58,76)(29,78,91,123)(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)(44,100,57,75)(49,104,68,113)(51,106,93,128)(53,107,74,62)(54,108,109,121)(60,111,88,114)(70,119,71,116)(73,118,92,126)(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)(20,69)(21,73)(22,70)(23,76)(25,85)(26,80)(27,82)(29,34)(30,37)(33,51)(36,94)(38,96)(39,81)(40,71)(42,99)(44,102)(45,90)(47,105)(48,108)(49,53)(52,109)(55,93)(56,89)(57,103)(59,111)(61,88)(62,113)(64,115)(65,118)(66,116)(67,119)(68,74)(72,92)(75,120)(77,106)(78,84)(79,86)(83,121)(87,124)(91,95)(97,125)(98,123)(100,122)(101,127)(104,107)(110,128)(112,114)(117,126);(2,13)(3,20)(11,47)(12,50)(14,52)(15,16)(18,64)(19,69)(21,72)(22,40)(23,56)(28,58)(29,91)(33,55)(34,95)(46,105)(48,83)(51,93)(54,109)(63,115)(65,117)(66,67)(70,71)(73,92)(76,89)(77,110)(78,123)(84,98)(106,128)(108,121)(116,119)(118,126)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,e^b=ei,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, 7530866);