Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388379 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14519 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1262 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)(3,13,16,8)(4,9,20,33)(5,10)(6,12)(7,11)(14,27)(15,28,43,34)(17,38,24,29)(18,30)(19,31,49,59)(21,32,51,61)(22,36)(23,35)(25,39)(26,37)(40,42)(41,62)(44,54,45,55)(46,64)(47,57)(48,56)(50,58,52,60)(53,63);(1,3)(2,8)(4,18)(5,15)(6,24)(7,16)(9,30)(10,34)(11,13)(12,38)(14,21)(17,26)(19,48)(20,47)(22,45)(23,43)(25,44)(27,32)(28,35)(29,37)(31,56)(33,57)(36,54)(39,55)(40,61)(41,50)(42,51)(46,49)(52,63)(53,60)(58,62)(59,64);(1,4)(2,9,11,33)(3,14,16,42)(5,19)(6,21)(7,20)(8,27)(10,31,35,59)(12,32,37,61)(13,40)(15,41,43,63)(17,18,24,47)(22,50)(23,49)(25,52)(26,51)(28,53)(29,30)(34,62)(36,58,39,60)(38,57)(44,48,45,46)(54,64)(55,56);(1,5,7,23)(2,10,11,35)(3,15,16,43)(4,19,20,49)(6,25,26,22)(8,28,13,34)(9,31,33,59)(12,39,37,36)(14,41,42,63)(17,45,24,44)(18,46,47,48)(21,52,51,50)(27,53,40,62)(29,55,38,54)(30,56,57,64)(32,60,61,58)Polycyclic presentation:
2^9|a^2=h,c^2=eh,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72285845, #72286106, #72286194, … |
| maximal normal subgroup of | #72285845, #72286106, #72286194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295576);