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 | #61208 | C2 |
| 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)(12,24)(13,27)(14,18)(15,20)(16,26)(17,21)(22,36)(23,32)(25,39)(28,38)(29,34)(30,35)(31,33)(37,40);(2,6)(7,8)(11,20)(12,25)(13,21)(15,19)(17,27)(22,37)(23,34)(24,39)(28,31)(29,32)(33,38)(36,40);(1,3,5,4)(2,7,6,8)(9,11,21,34,14,23,13,20)(10,15,27,32,18,29,17,19)(12,22,35,37,25,38,26,33)(16,28,39,40,30,36,24,31);(1,3,5,4)(2,7,6,8)(9,12,13,26,14,25,21,35)(10,16,17,24,18,30,27,39)(11,22,20,33,23,38,34,37)(15,28,19,31,29,36,32,40);(1,4,5,3)(2,7,6,8)(11,23)(15,29)(19,32)(20,34)(22,38)(28,36)(31,40)(33,37)Polycyclic presentation:
2^9|c^2=fgi,d^2=fg,e^2=f,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, 7472626);