Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #452727 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16061 |
| 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 | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,6,12,7,13)(3,9,18,39,19,33,17,32)(4,20,24,10,25,38,23,36)(5,26,28,40,27,11,29,37)(15,41,47,30,43,34,21,50)(16,44,45,60,46,31,48,59)(22,52,54,35,55,49,56,57)(42,61,51,58,63,62,53,64);(1,3)(2,9,12,33)(4,21)(5,16,27,46)(6,19)(7,17)(8,18)(10,34,36,41)(11,31)(13,32,14,39)(15,24)(20,30,38,50)(22,53,55,51)(23,43)(25,47)(26,44)(28,45,29,48)(35,62)(37,59)(40,60)(42,56,63,54)(49,64)(52,58)(57,61);(1,4,7,24,6,25,8,23)(2,10,13,38,12,36,14,20)(3,15,18,21,19,43,17,47)(5,22,28,56,27,55,29,54)(9,30,39,41,33,50,32,34)(11,35,37,52,26,57,40,49)(16,42,48,51,46,63,45,53)(31,58,60,62,44,64,59,61);(1,5)(2,11)(3,16)(4,22)(6,27)(7,28)(8,29)(9,31)(10,35)(12,26)(13,37)(14,40)(15,42)(17,45)(18,48)(19,46)(20,49)(21,51)(23,54)(24,56)(25,55)(30,58)(32,59)(33,44)(34,61)(36,57)(38,52)(39,60)(41,62)(43,63)(47,53)(50,64)Polycyclic presentation:
2^9|a^2=egh,b^2=efg,c^2=efi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359924);