Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509294 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30567 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,9,5,12)(3,15,19,31,18,43,30,10)(4,20,21,11,23,37,24,32)(6,13,26,42,29,33,25,39)(16,44,48,34,47,40,27,36)(17,35,52,45,50,59,51,61)(22,53,54,38,28,58,56,41)(46,62,55,63,49,60,57,64);(2,9)(3,16)(5,7)(6,26)(10,34)(13,39)(15,44)(17,49)(18,47)(19,27)(20,37)(21,24)(22,54)(25,29)(28,56)(30,48)(31,36)(33,42)(35,60)(38,58)(40,43)(41,53)(45,63)(46,50)(51,57)(52,55)(59,62)(61,64);(1,3,7,30,8,18,5,19)(2,10,12,15,9,31,14,43)(4,17,21,51,23,50,24,52)(6,27,26,47,29,48,25,16)(11,35,20,45,32,59,37,61)(13,40,39,36,33,44,42,34)(22,55,54,46,28,57,56,49)(38,63,53,60,41,64,58,62);(1,4)(2,11)(3,17)(5,24)(6,28)(7,21)(8,23)(9,32)(10,35)(12,20)(13,41)(14,37)(15,45)(16,46)(18,50)(19,52)(22,29)(25,54)(26,56)(27,57)(30,51)(31,59)(33,38)(34,60)(36,62)(39,58)(40,64)(42,53)(43,61)(44,63)(47,49)(48,55)Polycyclic presentation:
2^9|a^2=h,c^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=ghi,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416490);