Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #92825 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1270 |
| commutator | C2×C42 | C2×C8 |
| Frattini, cosocle | C2×#336 | C22 |
| prime radical | C24 | #100 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C2×#336 ▷ C23×C4 ▷ C22 ▷ C1 | |
415 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,18,14,19,11,21,15,20,22,16,13,17,23,24);(2,5)(6,7)(9,11,15,23)(10,16,20,18)(12,13,22,14)(17,24,19,21)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=f,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,g^b=gi,d^c=dhi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27);