Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
475 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,4,7,5,3,6,8)(9,10,13,15)(11,17,19,24)(12,20,21,16)(14,22,23,18);(9,11)(10,14)(12,18,21,22)(13,19)(15,23)(16,17,20,24);(1,3,4,8,5,2,6,7)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=i,e^2=h,b^a=bd,c^a=ci,d^a=di,f^a=fgi,d^b=dg,g^b=gi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35652);