Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1519 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C2 | C2×#1519 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
995 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)(3,14,18,41)(4,20,23,10)(5,11)(6,26,28,58)(7,30)(8,13)(9,17)(12,25,38,29)(15,42)(16,40,19,37)(21,50,24,35)(22,53)(27,32)(31,44,49,63)(33,47)(34,48)(36,52)(39,46)(43,61)(45,62)(51,55)(54,60,64,59)(56,57);(1,3)(2,9)(4,15)(5,16)(6,27)(7,19)(8,18)(10,31)(11,32)(12,37)(13,34)(14,26)(17,29)(20,49)(21,43)(22,54)(23,46)(24,45)(25,48)(28,47)(30,33)(35,59)(36,61)(38,40)(39,57)(41,58)(42,53)(44,55)(50,60)(51,62)(52,63)(56,64);(1,4)(2,10)(3,15)(5,21)(6,22)(7,24)(8,23)(9,31)(11,35)(12,36)(13,20)(14,39)(16,43)(17,44)(18,46)(19,45)(25,52)(26,57)(27,54)(28,56)(29,55)(30,50)(32,59)(33,60)(34,49)(37,61)(38,51)(40,62)(41,42)(47,64)(48,63)(53,58);(1,5,8,7)(2,11,13,30)(3,16,18,19)(4,21,23,24)(6,25,28,29)(9,32,34,33)(10,35,20,50)(12,26,38,58)(14,40,41,37)(15,43,46,45)(17,27,48,47)(22,52,56,55)(31,59,49,60)(36,57,51,53)(39,62,42,61)(44,54,63,64)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280870, #72280909, #72280935, … |
| maximal normal subgroup of | #72280870, #72280909, #72280935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389228);