Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497451 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22⋊C4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,15)(4,21,24,10)(5,26)(6,31)(7,37)(8,13)(9,18)(11,29)(12,35)(14,34)(16,59,62,56)(17,64)(19,71)(20,58)(22,77,80,91)(23,82,86,97)(25,46,85,76)(27,92)(28,95)(30,54)(32,53)(33,51)(36,55)(38,50)(39,49)(40,61,73,117)(41,94)(42,70)(43,93)(44,81,74,89)(45,87,75,115)(47,67)(48,65)(52,68)(57,66)(60,103,113,119)(63,105,88,111)(69,101)(72,90)(78,99,124,120)(79,104,112,121)(83,100,123,107)(84,122,126,106)(96,102)(98,108)(109,114,110,118)(116,128,125,127);(1,3,8,20)(2,9,13,42)(4,16,24,62)(5,27,30,94)(6,32,36,93)(7,38,34,19)(10,40,21,73)(11,47,49,64)(12,50,52,71)(14,53,37,43)(15,55,58,31)(17,65,67,28)(18,68,70,35)(22,78,80,124)(23,83,86,123)(25,88,85,63)(26,90,54,57)(29,96,39,69)(33,72,98,66)(41,95,92,48)(44,103,74,119)(45,105,75,111)(46,107,76,100)(51,102,108,101)(56,82,59,97)(60,112,113,79)(61,115,117,87)(77,109,91,110)(81,125,89,116)(84,118,126,114)(99,121,120,104)(106,128,122,127);(1,4)(2,10)(3,16)(5,22)(6,23)(7,25)(8,24)(9,40)(11,44)(12,45)(13,21)(14,46)(15,56)(17,60)(18,61)(19,63)(20,62)(26,91)(27,78)(28,79)(29,81)(30,80)(31,97)(32,83)(33,84)(34,85)(35,87)(36,86)(37,76)(38,88)(39,89)(41,99)(42,73)(43,100)(47,103)(48,104)(49,74)(50,105)(51,106)(52,75)(53,107)(54,77)(55,82)(57,109)(58,59)(64,119)(65,112)(66,114)(67,113)(68,115)(69,116)(70,117)(71,111)(72,118)(90,110)(92,120)(93,123)(94,124)(95,121)(96,125)(98,126)(101,127)(102,128)(108,122);(1,5,8,30)(2,11,13,49)(3,17,20,67)(4,22,24,80)(6,33,36,98)(7,39,34,29)(9,41,42,92)(10,44,21,74)(12,51,52,108)(14,54,37,26)(15,57,58,90)(16,60,62,113)(18,69,70,96)(19,72,38,66)(23,84,86,126)(25,89,85,81)(27,93,94,32)(28,68,65,35)(31,48,55,95)(40,99,73,120)(43,102,53,101)(45,106,75,122)(46,77,76,91)(47,71,64,50)(56,109,59,110)(61,116,117,125)(63,118,88,114)(78,123,124,83)(79,115,112,87)(82,121,97,104)(100,128,107,127)(103,111,119,105)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=eh,h^b=hi,g^c=gi,e^d=eh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404649);