Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125287 |
| 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 | C2×#466 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(2,6,7,8)(9,10,13,19,16,21,23,24,14,22,11,17,15,20,12,18);(1,2)(3,6)(4,8)(5,7)(10,17)(11,12)(13,23)(15,16)(18,20)(19,22)(21,24);(1,3,5,4)(2,6,7,8)(9,11,15,23,14,13,16,12)(10,17,20,24,22,19,21,18)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32483);