Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682628 |
| 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 | |
1840 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)(3,6)(4,5)(7,8)(9,10,15,11)(12,13,18,16)(14,17,23,19)(20,21,22,24);(2,5)(3,7)(9,11)(10,15)(12,16)(13,18)(14,22)(17,21)(19,24)(20,23);(9,12,16,10,15,18,13,11)(14,20,24,17,23,22,21,19);(2,5)(6,8)(9,13,15,16)(10,12,11,18)(14,21,23,24)(17,20,19,22);(1,3,4,7)(2,6,5,8)(9,14,15,23)(10,17,11,19)(12,20,18,22)(13,21,16,24)Polycyclic presentation:
2^9|a^2=i,c^2=g,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,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592695);