Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1964 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1964 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,5,11)(3,15,18,42,20,44,29,43)(4,21,24,39,25,37,22,10)(6,26,16,40,28,12,30,32)(9,31,35,27,36,50,34,19)(17,46,47,41,49,62,56,63)(23,53,45,58,55,52,57,38)(33,48,51,59,61,54,60,64);(1,3,16,31)(2,9,32,42)(4,17,45,59)(5,18,28,19)(6,27,7,29)(8,20,30,50)(10,33,53,62)(11,34,26,44)(12,15,13,35)(14,36,40,43)(21,51,38,41)(22,47,55,48)(23,54,24,56)(25,49,57,64)(37,60,58,63)(39,61,52,46);(1,4,5,22,8,25,7,24)(2,10,13,21,14,39,11,37)(3,17,18,47,20,49,29,56)(6,23,16,45,28,55,30,57)(9,33,35,51,36,61,34,60)(12,38,40,52,26,58,32,53)(15,41,43,46,44,63,42,62)(19,48,50,64,27,54,31,59);(1,5,8,7)(2,11,14,13)(3,18,20,29)(4,22,25,24)(6,16,28,30)(9,34,36,35)(10,37,39,21)(12,32,26,40)(15,42,44,43)(17,47,49,56)(19,50,27,31)(23,45,55,57)(33,60,61,51)(38,53,58,52)(41,62,63,46)(48,64,54,59)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280870, #72281046, #72281073, … |
| maximal normal subgroup of | #72280870, #72281046, #72281073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402398);