Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401788 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14823 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,18,40,19,9,30,33)(4,20,24,10,25,37,23,36)(5,26,28,38,29,11,27,39)(16,43,21,31,45,34,48,41)(17,32,46,58,47,42,56,60)(22,51,53,55,54,49,52,35)(44,59,50,61,63,62,64,57);(1,3)(2,9)(4,21)(5,17)(6,18)(7,30)(8,19)(10,34)(11,32)(12,15)(13,33)(14,40)(16,24)(20,41)(22,50)(23,45)(25,48)(26,42)(27,46)(28,56)(29,47)(31,37)(35,59)(36,43)(38,58)(39,60)(44,53)(49,61)(51,57)(52,63)(54,64)(55,62);(1,4,8,25)(2,10,12,36)(3,16,19,45)(5,22,29,54)(6,24,7,23)(9,31,15,41)(11,35,26,55)(13,37,14,20)(17,44,47,63)(18,48,30,21)(27,53,28,52)(32,57,42,61)(33,43,40,34)(38,51,39,49)(46,64,56,50)(58,62,60,59);(1,5)(2,11,12,26)(3,17)(4,22)(6,27)(7,28)(8,29)(9,32,15,42)(10,35,36,55)(13,38,14,39)(16,44)(18,46)(19,47)(20,49,37,51)(21,50)(23,52)(24,53)(25,54)(30,56)(31,57,41,61)(33,58,40,60)(34,59,43,62)(45,63)(48,64)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308985);