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 | |
410 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 410 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,3,7,8)(9,10)(11,18,20,23)(12,21)(13,16)(14,17)(15,19,24,22)(25,26)(27,31,34,38)(28,35,36,30)(29,32)(33,39)(37,40);(1,3)(2,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(26,30,32,35)(28,36)(31,37,38,39)(33,40);(2,3)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,27,29,34)(26,31,32,38)(28,33,36,40)(30,37,35,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=h,b^a=bd,c^a=cfg,d^a=df,f^a=fi,c^b=ch,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, 32816);