Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,25)(14,29)(15,19)(16,23)(18,27)(21,36)(22,37)(24,35)(26,32)(28,39)(30,34)(31,38)(33,40);(1,3)(2,5)(4,8)(6,7)(11,21)(16,31)(20,36)(22,30)(23,38)(26,40)(32,33)(34,37);(3,8)(5,7)(9,11)(10,16)(12,22)(13,26)(14,30)(15,21)(17,32)(18,34)(19,31)(20,25)(23,27)(24,38)(28,40)(29,33)(35,37)(36,39);(2,6)(5,7);(9,12,15,14)(10,17,19,29)(11,22,21,30)(13,24,28,27)(16,32,31,33)(18,25,35,39)(20,37,36,34)(23,26,38,40)Polycyclic presentation:
2^9|e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cfi,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308542 |
| maximal normal subgroup of | #72308542 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530492);