Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392727 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14694 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,14)(3,15,20,32,19,38,18,9)(4,21,24,37,25,10,23,34)(5,26,28,11,29,35,27,36)(16,39,44,30,43,56,42,48)(17,31,47,57,46,58,45,40)(22,50,52,33,53,54,51,49)(41,55,62,59,61,63,60,64);(1,3)(2,9,13,32)(4,16)(5,17)(6,18)(7,20)(8,19)(10,30,21,48)(11,31,36,58)(12,15,14,38)(22,41)(23,42)(24,44)(25,43)(26,40,35,57)(27,45)(28,47)(29,46)(33,55,49,63)(34,56,37,39)(50,64,54,59)(51,60)(52,62)(53,61);(1,4)(2,10)(3,16)(5,22)(6,24)(7,23)(8,25)(9,30)(11,33)(12,37)(13,21)(14,34)(15,39)(17,41)(18,44)(19,43)(20,42)(26,54)(27,52)(28,51)(29,53)(31,55)(32,48)(35,50)(36,49)(38,56)(40,59)(45,62)(46,61)(47,60)(57,64)(58,63);(1,5,8,29)(2,11,13,36)(3,17,19,46)(4,22,25,53)(6,28,7,27)(9,31,32,58)(10,33,21,49)(12,26,14,35)(15,40,38,57)(16,41,43,61)(18,47,20,45)(23,52,24,51)(30,55,48,63)(34,50,37,54)(39,59,56,64)(42,62,44,60)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72306351, #72306448, #72306470, … |
| maximal normal subgroup of | #72306351, #72306448, #72306470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299926);