Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| 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 | |
2088 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)(3,7)(4,8)(5,6)(9,10)(11,20)(12,23)(13,26)(14,18)(15,31)(16,22)(17,30)(19,28)(21,36)(24,33)(25,35)(27,32)(29,40)(34,39)(37,38);(2,6)(7,8)(9,11,14,22)(10,16,18,20)(12,15,25,28)(13,21,29,37)(17,33,34,32)(19,35,31,23)(24,30,27,39)(26,38,40,36);(1,3,5,4)(2,7,6,8)(9,12,15,22,14,25,28,11)(10,16,19,23,18,20,31,35)(13,27,30,21,29,24,39,37)(17,33,26,36,34,32,40,38);(1,4,5,3)(2,8,6,7)(9,13,28,39,14,29,15,30)(10,17,31,40,18,34,19,26)(11,21,25,27,22,37,12,24)(16,32,35,36,20,33,23,38);(2,6)(7,8)(10,18)(16,20)(17,34)(19,31)(23,35)(26,40)(32,33)(36,38)Polycyclic presentation:
2^9|b^2=i,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,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469007);