Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471338 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16788 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,6,11,14,12)(3,8,29,32,17,34,18,33)(4,19,23,37,22,9,24,41)(5,25,27,10,26,40,28,39)(15,30,44,35,43,58,20,50)(16,45,49,59,47,60,48,31)(21,52,54,36,53,56,55,38)(42,61,51,57,46,62,63,64);(1,3,14,29,6,17,7,18)(2,8,13,33,11,34,12,32)(4,20,23,43,22,44,24,15)(5,16,27,48,26,47,28,49)(9,35,37,58,19,50,41,30)(10,31,25,45,39,59,40,60)(21,51,55,46,53,63,54,42)(36,61,56,64,38,62,52,57);(1,4,6,22)(2,9,11,19)(3,15)(5,21,26,53)(7,23,14,24)(8,30)(10,38,39,36)(12,37,13,41)(16,46)(17,43)(18,20)(25,56,40,52)(27,54,28,55)(29,44)(31,57)(32,35)(33,50)(34,58)(42,47)(45,62)(48,51)(49,63)(59,64)(60,61);(1,5)(2,10,11,39)(3,16)(4,21)(6,26)(7,28)(8,31,34,59)(9,36,19,38)(12,40,13,25)(14,27)(15,42)(17,47)(18,49)(20,51)(22,53)(23,55)(24,54)(29,48)(30,57,58,64)(32,60,33,45)(35,61,50,62)(37,56,41,52)(43,46)(44,63)Polycyclic presentation:
2^9|a^2=eg,b^2=efh,c^2=f,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378535);