Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125393 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×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 | |
446 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,4,6,5,8,3,7)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23)(25,26)(27,33)(28,36,37,30)(29,32)(31,34)(35,38,40,39);(2,6)(3,4)(7,8)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21)(26,30,32,36)(27,34)(28,37)(31,39,33,38);(1,3,5,4)(2,6,8,7)(10,15)(14,22)(16,23)(17,24)(25,27,29,34)(26,31,32,33)(28,35,37,40)(30,38,36,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=hi,d^2=gi,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fi,c^b=chi,d^b=dg,e^b=efgi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32654);