Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814805 |
| 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 | |
1864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,11)(12,18,15,16)(13,17,23,19)(20,21,22,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,21)(15,16)(17,20)(19,22)(23,24);(10,16,11,18)(17,24,19,21);(2,6)(5,7)(9,12,14,15)(10,16,11,18)(13,22,23,20)(17,21,19,24);(3,8)(5,7)(9,13,12,20,14,23,15,22)(10,17,18,21,11,19,16,24)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722005);