Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14375 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
996 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)(3,8)(4,9,19,35)(5,21,23,10)(6,24)(7,11)(12,28)(13,29,42,56)(14,43,45,30)(15,38)(16,31)(17,48,46,33)(18,50)(20,39,26,36)(22,37,27,40)(25,32)(34,52)(41,63)(44,57,47,58)(49,59)(51,60)(53,61)(54,62)(55,64);(1,3,7,16)(2,8,11,31)(4,17,19,46)(5,14,23,45)(6,25,28,15)(9,33,35,48)(10,30,21,43)(12,38,24,32)(13,26,42,20)(18,51,52,49)(22,47,27,44)(29,39,56,36)(34,60,50,59)(37,58,40,57)(41,54,64,53)(55,62,63,61);(1,4)(2,9,11,35)(3,13)(5,18)(6,26)(7,19)(8,29,31,56)(10,34,21,50)(12,39,24,36)(14,41)(15,46)(16,42)(17,25)(20,28)(22,54)(23,52)(27,53)(30,55,43,63)(32,48,38,33)(37,62,40,61)(44,49)(45,64)(47,51)(57,59,58,60);(1,5)(2,10)(3,14)(4,18)(6,27)(7,23)(8,30)(9,34)(11,21)(12,40)(13,41)(15,47)(16,45)(17,49)(19,52)(20,54)(22,28)(24,37)(25,44)(26,53)(29,55)(31,43)(32,58)(33,59)(35,50)(36,62)(38,57)(39,61)(42,64)(46,51)(48,60)(56,63)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291449);