Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #440904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15670 |
| 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 | |
916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,29,8,12,7,13)(3,14,30,34,17,9,18,33)(4,19,22,37,24,10,23,40)(5,25,28,39,26,56,27,11)(15,41,47,48,43,31,20,35)(16,44,45,32,46,55,57,38)(21,51,54,36,52,61,53,49)(42,59,50,58,63,60,64,62);(1,3,6,18,8,17,7,30)(2,9,29,34,12,14,13,33)(4,20,22,43,24,47,23,15)(5,16,27,45,26,46,28,57)(10,35,40,31,19,48,37,41)(11,38,56,44,39,32,25,55)(21,50,53,63,52,64,54,42)(36,60,51,62,49,59,61,58);(1,4,8,24)(2,10,12,19)(3,15,17,43)(5,21,26,52)(6,23,7,22)(9,31,14,41)(11,36,39,49)(13,40,29,37)(16,42,46,63)(18,47,30,20)(25,51,56,61)(27,54,28,53)(32,58,38,62)(33,48,34,35)(44,60,55,59)(45,64,57,50);(1,5)(2,11,12,39)(3,16)(4,21)(6,27)(7,28)(8,26)(9,32,14,38)(10,36,19,49)(13,25,29,56)(15,42)(17,46)(18,45)(20,50)(22,53)(23,54)(24,52)(30,57)(31,58,41,62)(33,44,34,55)(35,59,48,60)(37,61,40,51)(43,63)(47,64)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348101);