Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | #61097 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,15,20)(11,21,25,16)(12,13,28,31)(14,32,29,19)(17,36,30,18)(22,39,33,37)(23,27,38,40)(24,34,26,35);(2,6)(7,8)(9,11,15,25)(10,16,20,21)(12,27,28,40)(13,23,31,38)(14,24,29,26)(17,37,30,39)(18,22,36,33)(19,34,32,35);(1,3,5,4)(2,7,6,8)(9,12,14,18,15,28,29,36)(10,17,19,13,20,30,32,31)(11,22,26,27,25,33,24,40)(16,23,35,37,21,38,34,39);(1,3,5,4)(2,7,6,8)(9,13,29,17,15,31,14,30)(10,18,32,12,20,36,19,28)(11,23,24,39,25,38,26,37)(16,33,34,27,21,22,35,40);(1,4,5,3)(2,7,6,8)(9,14,15,29)(10,19,20,32)(11,24,25,26)(12,18,28,36)(13,30,31,17)(16,34,21,35)(22,40,33,27)(23,39,38,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,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,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472868);