Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,23)(13,26)(14,29)(15,19)(17,27)(18,24)(20,35)(21,34)(22,31)(25,32)(28,39)(30,36)(33,38)(37,40);(2,5)(6,8)(9,11,15,22)(10,16,19,31)(12,24,21,14)(13,20,28,37)(17,32,33,30)(18,34,29,23)(25,38,36,27)(26,35,39,40);(1,3,4,7)(2,6,5,8)(9,12,14,11,15,21,24,22)(10,16,18,23,19,31,29,34)(13,25,27,20,28,36,38,37)(17,30,26,35,33,32,39,40);(9,13,24,38,15,28,14,27)(10,17,29,39,19,33,18,26)(11,20,12,25,22,37,21,36)(16,30,34,40,31,32,23,35);(2,5)(6,8)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=gi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467885);