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 |
| prime radical | #61209 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1706 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)(12,24)(13,15)(14,27)(16,20)(17,26)(18,21)(22,36)(23,30)(25,39)(28,34)(29,35)(31,38)(32,33)(37,40);(2,6)(7,8)(10,15)(11,20)(12,25)(14,21)(16,30)(17,29)(19,28)(22,37)(23,34)(31,40)(32,36)(33,38);(1,3,5,4)(2,7,6,8)(9,11,21,34,13,23,14,20)(10,16,27,30,15,28,18,19)(12,22,35,37,25,38,26,33)(17,31,39,40,29,36,24,32);(1,3,5,4)(2,7,6,8)(9,12,14,26,13,25,21,35)(10,17,18,24,15,29,27,39)(11,22,20,33,23,38,34,37)(16,31,19,32,28,36,30,40);(1,4,5,3)(2,7,6,8)(10,15)(11,23)(17,29)(18,27)(20,34)(22,38)(24,39)(33,37)Polycyclic presentation:
2^9|c^2=fg,d^2=fgi,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, 7472625);