Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C2×#537 | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
897 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 897 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,8)(9,10,13,18,14,17,21,24)(11,15,19,22,12,16,20,23);(1,2)(3,5)(4,6)(7,8)(10,15)(11,12)(16,17)(18,23)(19,20)(22,24);(1,3)(2,5)(4,7)(6,8)(9,11,14,12)(10,16,17,15)(13,20,21,19)(18,22,24,23)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgi,d^c=dh,f^c=fi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72303863 |
| maximal normal subgroup of | #72303863 |
| maximal quotient of | #72303887 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40520);