Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125064 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,19,21,15)(12,17)(13,23)(14,18)(16,22,24,20);(1,3,6,2)(4,7,5,8)(10,15,18,19)(11,20)(12,13)(16,17,24,23)(21,22);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^9|a^2=e,b^2=fg,c^2=h,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efg,f^b=fi,g^b=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32410);