Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,8)(10,14,15,22)(11,17)(12,20)(16,23,21,18)(19,24);(1,2)(3,5)(4,6)(7,8)(9,10,12,21,13,15,20,16)(11,18,17,14,19,23,24,22);(1,3)(2,5)(4,7)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219622, #72224575, #72224582, … |
| maximal normal subgroup of | #72219622, #72224575, #72224582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43204);