Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7515 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1402 |
| commutator | #127 | C2×C8 |
| Frattini, cosocle | #1957 | C22 |
| prime radical | C2×D42 | C4 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1957 ◁ G | |
| LECS | G ▷ #1957 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
415 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 28 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 — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,18,19,14)(12,16,22,15)(13,17)(20,23,21,24);(10,14)(15,23)(16,18)(17,24)Polycyclic presentation:
2^9|a^2=d,d^2=f,e^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=dehi,e^b=ehi,g^b=ghi,d^c=dgh,e^c=ehi,g^c=gi,e^d=ehi,g^d=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72280029, #72280041, #72280066 |
| maximal normal subgroup of | #72280029, #72280041, #72280066 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1734);