Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | |
1793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,21)(12,25)(13,20)(14,18)(15,19)(16,24)(17,26)(22,32)(23,38)(27,35)(28,37)(29,40)(30,36)(31,34)(33,39);(2,5)(3,7)(9,11,15,24)(10,16,19,21)(12,26,22,13)(14,29,30,27)(17,25,20,32)(18,35,36,40)(23,31,39,28)(33,34,38,37);(9,12,26,11,15,22,13,24)(10,16,20,32,19,21,17,25)(14,27,31,23,30,29,28,39)(18,33,37,40,36,38,34,35);(2,5)(6,8)(9,13,15,26)(10,17,19,20)(11,22,24,12)(14,28,30,31)(16,32,21,25)(18,34,36,37)(23,29,39,27)(33,40,38,35);(1,3,4,7)(2,6,5,8)(9,14,15,30)(10,18,19,36)(11,23,24,39)(12,27,22,29)(13,28,26,31)(16,33,21,38)(17,34,20,37)(25,35,32,40)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592727);