Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184811 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(17,18)(19,23,24,21)(20,22);(2,5)(6,7)(9,10,12,14)(11,15,16,13);(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,24)(22,23);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72314076, #72314085, #72314093 |
| maximal normal subgroup of | #72314076, #72314085, #72314093 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270093);