Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1862 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1862 conjugacy classes of subgroups — too many to draw. Only its 206 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)(3,7,8,5)(4,6)(10,15,17,20)(16,23,24,22);(1,3)(2,5)(4,8)(6,7)(9,10,13,15,14,17,12,20)(11,16,18,22,19,24,21,23);(9,11)(10,16)(12,21)(13,18)(14,19)(15,22)(17,24)(20,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ei,f^b=fhi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214184, #72214239, #72214258, … |
| maximal normal subgroup of | #72214184, #72214239, #72214258, … |
| maximal quotient of | C2×#107135 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14331);