Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137149 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#1999 | C2 |
| 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 | |
922 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 922 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:
(1,2)(3,7,8,5)(4,6)(9,10,13,17,23,21,12,22,15,18,24,20,14,11,19,16);(1,2)(3,5)(4,6)(7,8)(9,11,14,10,15,21,23,18)(12,22,13,20,19,16,24,17);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219631, #72224575, #72224582, … |
| maximal normal subgroup of | #72219631, #72224575, #72224582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44348);