Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11836 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1514 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,5)(4,6)(7,8)(9,10,12,19,15,17,23,24,13,21,22,16,14,20,11,18);(2,5)(3,8)(6,7)(10,16)(11,22)(12,23)(14,15)(17,24)(18,20)(19,21);(2,6)(5,7)(10,17,21,20)(11,23)(12,22)(16,24,19,18)Polycyclic presentation:
2^9|a^2=e,c^2=fghi,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240792, #72240794, #72240796, … |
| maximal normal subgroup of | #72240792, #72240794, #72240796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31830);