Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125393 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | C25 | C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 528 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,18,12,16)(11,15,19,22,20,23,21,24)(25,26)(27,35,37,40)(28,33,29,32)(30,34)(31,36)(38,39);(2,5)(3,8)(6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,22)(18,23)(25,27)(26,31,34,39)(28,38)(29,36)(30,37)(32,40,33,35);(9,12,14,13)(10,16,18,17)(11,19,20,21)(15,22,23,24)(25,28,30,29)(26,32,34,33)(27,36,37,38)(31,35,39,40)Polycyclic presentation:
2^9|a^2=e,b^2=fg,c^2=hi,d^2=gi,e^2=h,b^a=bd,c^a=cfg,d^a=df,f^a=fi,c^b=chi,d^b=dg,e^b=efgi,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, 32797);