Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132185 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #42969 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#550 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
887 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 887 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(2,5,6,7)(9,10)(11,18,19,22)(12,16,21,15)(13,17)(14,20)(23,24);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,17,23)(12,20,21,24)(15,22,16,18);(3,8)(5,7)(10,15)(11,19)(14,18)(16,17)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fgi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39415);