Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16152 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,30,34,19,33,18,9)(4,20,23,10,25,42,24,37)(5,11,28,39,29,41,27,40)(16,31,45,35,46,58,21,43)(17,47,56,32,48,38,26,54)(22,49,51,36,53,62,52,60)(44,63,55,61,64,57,50,59);(1,3,8,19)(2,9)(4,21,25,45)(5,26)(6,18,7,30)(10,35)(11,38,41,47)(12,15)(13,33)(14,34)(16,24,46,23)(17,28)(20,31)(22,44)(27,48)(29,56)(32,39,54,40)(36,57,60,63)(37,43)(42,58)(49,61,62,59)(50,52)(51,55)(53,64);(1,4,6,23,8,25,7,24)(2,10,12,20,14,37,13,42)(3,16,18,45,19,46,30,21)(5,22,27,51,29,53,28,52)(9,31,15,43,34,58,33,35)(11,36,39,49,41,60,40,62)(17,44,26,55,48,64,56,50)(32,57,38,61,54,63,47,59);(1,5,8,29)(2,11)(3,17)(4,22,25,53)(6,27,7,28)(9,32,34,54)(10,36)(12,39)(13,40)(14,41)(15,38,33,47)(16,44)(18,26)(19,48)(20,49)(21,50)(23,51,24,52)(30,56)(31,57,58,63)(35,59,43,61)(37,60)(42,62)(45,55)(46,64)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,c^2=e,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361045);