Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
371 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 132 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:
(1,2,4,7,5,3,6,8)(10,15,17,20)(11,12)(13,23)(16,24,19,18)(21,22);(9,10,12,18,23,16,11,20,14,17,22,24,13,19,21,15);(1,3,4,8,5,2,6,7)(10,16,17,19)(11,21)(12,22)(15,24,20,18)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=cg,d^b=di,e^b=ei,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216353, #72216443, #72216451, … |
| maximal normal subgroup of | #72216353, #72216443, #72216451, … |
| maximal quotient of | #72216346, #72223574, C2×#128692 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35888);