Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142624 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C23×C8 | C23 |
| prime radical | #25779 | C2 |
| LCS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
720 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 720 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18);(5,8)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22);(1,3)(2,5)(4,7)(6,8)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,g^c=gi,h^c=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72280142 |
| maximal normal subgroup of | #72280142 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49824);