Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
770 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 770 conjugacy classes of subgroups — too many to draw. Only its 56 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)(3,5)(4,6)(7,8)(9,10,11,17)(12,16,18,24)(13,15,19,23)(14,20,22,21);(2,5)(3,8)(6,7)(10,14)(11,18)(12,21)(15,16)(19,20)(22,24);(2,6)(5,7)(10,15)(11,19)(14,16)(18,20)Polycyclic presentation:
2^9|a^2=e,d^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efgi,f^b=fi,e^c=ei,e^d=ehi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72213278, #72213311, #72213323, … |
| maximal normal subgroup of | #72213278, #72213311, #72213323, … |
| maximal quotient of | C2×#150564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57760);