Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
2108 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)(4,8)(5,6)(9,10)(11,19)(12,23)(13,27)(14,18)(15,20)(16,26)(17,28)(21,31)(22,34)(24,39)(25,29)(30,36)(32,38)(33,35)(37,40);(2,6)(7,8)(11,20)(12,24)(13,28)(15,19)(17,27)(21,37)(22,36)(23,39)(25,35)(29,33)(30,34)(31,40);(1,3,5,4)(2,7,6,8)(9,11,13,20,14,22,28,36)(10,15,17,19,18,30,27,34)(12,25,26,37,24,21,38,35)(16,31,23,40,32,29,39,33);(1,4,5,3)(2,8,6,7)(9,12,13,26,14,24,28,38)(10,16,17,23,18,32,27,39)(11,21,20,35,22,25,36,37)(15,29,19,33,30,31,34,40);(2,6)(7,8)(11,22)(15,30)(19,34)(20,36)(21,25)(29,31)(33,40)(35,37)Polycyclic presentation:
2^9|c^2=fg,d^2=fg,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,d^c=di,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469011);