Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128808 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1762 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 102 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)(3,7)(4,6)(5,8)(10,15,17,24)(11,19)(12,13)(16,23,18,20)(21,22);(1,3)(2,5,6,7)(4,8)(9,10)(11,20)(12,16)(13,18)(14,17)(15,22)(19,24)(21,23);(10,16,17,18)(11,21)(15,23,24,20)(19,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280248, C2×#128808 |
| maximal normal subgroup of | #72280248, C2×#128808 |
| maximal quotient of | C2×#128808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36004);