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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 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)(11,19)(12,23)(13,15)(14,27)(16,20)(17,26)(18,25)(21,35)(22,30)(24,38)(28,34)(29,39)(31,37)(32,33)(36,40);(2,6)(7,8)(10,15)(11,20)(12,24)(14,25)(16,30)(17,29)(19,28)(21,36)(22,34)(31,40)(32,35)(33,37);(1,3,5,4)(2,7,6,8)(9,11,14,20,13,22,25,34)(10,16,18,19,15,28,27,30)(12,21,26,33,24,37,39,36)(17,31,23,32,29,35,38,40);(1,3,5,4)(2,7,6,8)(9,12,25,39,13,24,14,26)(10,17,27,38,15,29,18,23)(11,21,34,36,22,37,20,33)(16,31,30,40,28,35,19,32);(1,4,5,3)(2,7,6,8)(10,15)(11,22)(17,29)(18,27)(20,34)(21,37)(23,38)(33,36)Polycyclic presentation:
2^9|c^2=fgi,d^2=fg,e^2=fi,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, 7472627);