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 | |
2091 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)(11,16)(12,23)(13,25)(14,29)(15,18)(17,27)(19,33)(20,30)(21,28)(22,32)(24,31)(26,37)(34,35)(36,40)(38,39);(2,6)(7,8)(9,11,15,22)(10,16,18,32)(12,24,26,39)(13,28,29,20)(14,30,25,21)(17,33,34,40)(19,35,36,27)(23,31,37,38);(1,3,5,4)(2,7,6,8)(9,12,25,35,15,26,14,27)(10,17,29,37,18,34,13,23)(11,19,30,39,22,36,21,24)(16,31,28,40,32,38,20,33);(1,4,5,3)(2,8,6,7)(9,13,25,10,15,29,14,18)(11,20,30,16,22,28,21,32)(12,23,35,17,26,37,27,34)(19,33,39,31,36,40,24,38);(2,6)(7,8)(12,26)(17,34)(19,36)(23,37)(24,39)(27,35)(31,38)(33,40)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fgi,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,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, 7468993);