Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115302 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1886 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1886 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,7,8,5)(4,6)(9,10)(11,17)(12,20)(13,16)(14,18)(15,22)(19,24)(21,23);(5,7)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,3)(2,5)(4,8)(6,7)(12,22)(14,23)(15,20)(18,21)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=dh,d^c=di,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72267268, #72267306, #72267312 |
| maximal normal subgroup of | #72267268, #72267306, #72267312 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22500);