Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14811 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,16)(11,14,15,12)(18,22)(19,20);(5,6,7,8)(9,11,13,15)(10,14,16,12)(17,18)(19,24)(20,23)(21,22);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,20)(18,23,22,24);(1,2)(3,4)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22);(9,12,13,14)(10,15,16,11)(17,20,21,19)(18,23,22,24)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280660, #72280783 |
| maximal normal subgroup of | #72280660, #72280783 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1166746);