Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11989 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1551 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1551 |
| prime radical | #60682 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,25,12,40,38,7,13,27,39,5,10,24,36)(3,14,17,42,16,8,26,30,19,29,48,37,18,31,47,56)(4,20,23,49,50,35,58,61,22,33,51,59,21,9,32,34)(15,28,45,54,43,53,52,60,44,55,57,64,46,41,62,63);(1,3)(2,8)(4,15)(5,16)(6,26)(7,19)(9,28)(10,29)(11,37)(12,14)(13,31)(17,40)(18,25)(20,41)(21,43)(22,44)(23,52)(24,48)(27,47)(30,38)(32,57)(33,53)(34,60)(35,55)(36,56)(39,42)(45,58)(46,50)(49,54)(51,62)(59,64)(61,63);(1,4)(2,9,13,35)(3,15)(5,21)(6,23)(7,22)(8,28,31,55)(10,33,12,20)(11,34,39,61)(14,41,29,53)(16,43)(17,45)(18,46)(19,44)(24,32)(25,50)(26,52)(27,51)(30,54,56,64)(36,59,38,49)(37,60,42,63)(40,58)(47,62)(48,57);(1,5,7,25)(2,10,13,12)(3,16,19,18)(4,21,22,50)(6,24,27,40)(8,29,31,14)(9,33,35,20)(11,36,39,38)(15,43,44,46)(17,26,48,47)(23,32,51,58)(28,53,55,41)(30,37,56,42)(34,59,61,49)(45,52,57,62)(54,60,64,63)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=i,e^2=h,f^2=ghi,g^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390435);