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 | |
2129 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)(12,24)(13,28)(14,18)(15,20)(16,27)(17,21)(22,31)(23,34)(25,39)(26,29)(30,36)(32,37)(33,35)(38,40);(2,6)(7,8)(11,20)(12,25)(13,21)(15,19)(17,28)(22,38)(23,36)(24,39)(26,35)(29,33)(30,34)(31,40);(1,3,5,4)(2,7,6,8)(9,11,21,36,14,23,13,20)(10,15,28,34,18,30,17,19)(12,26,37,35,25,22,27,38)(16,31,39,33,32,29,24,40);(1,4,5,3)(2,8,6,7)(9,12,13,27,14,25,21,37)(10,16,17,24,18,32,28,39)(11,22,20,35,23,26,36,38)(15,29,19,33,30,31,34,40);(2,6)(7,8)(11,23)(15,30)(19,34)(20,36)(22,26)(29,31)(33,40)(35,38)Polycyclic presentation:
2^9|c^2=fgi,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, 7469015);