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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1784 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1784 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)(11,19)(12,16)(13,22)(14,18)(15,20)(17,23)(21,24);(5,7)(9,11,14,12)(10,15,18,24)(13,21,23,20)(16,22,19,17);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,16,18,19)(13,20,23,21)(15,17,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^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, #72267308, #72267319 |
| maximal normal subgroup of | #72267268, #72267308, #72267319 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22501);