Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125068 |
| 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 | |
422 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,6,7,8)(9,10,13,17,14,18,12,16)(11,19,22,15,20,24,21,23)(25,26)(27,33)(28,36,38,31)(29,32)(30,35,39,37)(34,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,22)(13,21)(14,20)(16,24)(17,19)(18,23)(25,27)(26,30,32,39)(28,37)(29,34)(31,40,36,33)(35,38);(1,3,5,4)(2,6,7,8)(11,20)(15,23)(19,24)(21,22)(27,34)(30,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,e^2=h,b^a=bd,d^a=df,f^a=fi,c^b=chi,d^b=dg,e^b=efgi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32438);