Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #3050 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 98 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)(10,15,16,23,18,24,17,21);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,21)(13,16)(14,18)(17,22)(19,23)(20,24);(2,6)(5,7)(9,11,13,19,14,20,22,12)(10,15,16,23,18,24,17,21)Polycyclic presentation:
2^9|a^2=e,c^2=gh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72219595, #72219596, #72219598, … |
| maximal normal subgroup of | #72219595, #72219596, #72219598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31616);