Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125393 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×#95 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,6,3,8)(9,10,12,14,13,11,15,16)(17,18)(19,23,27,32)(20,28,30,24)(21,25)(22,26)(29,31);(2,6)(3,4)(9,11)(10,13)(12,14)(15,16)(17,19,21,27)(18,22,25,31)(20,29,30,26)(23,24,32,28);(1,3,5,4)(2,7,6,8)(10,11)(14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,31)(25,32)(29,30)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=ghi,d^2=g,e^2=h,b^a=bd,c^a=cfg,d^a=df,f^a=fi,c^b=chi,d^b=dg,e^b=efg,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32813);