Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
670 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 198 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:
(1,2,4,7,5,8,3,6)(9,10)(11,19,21,31)(12,23,24,16)(13,26)(14,18)(15,20)(17,27)(22,36,35,32)(25,30,38,37)(28,40)(29,34)(33,39);(9,11,13,22,14,21,27,35)(10,15,17,28,18,29,26,39)(12,20,25,33,24,34,38,40)(16,19,30,32,23,31,37,36);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,f^2=i,b^a=bd,d^a=dg,f^a=fi,g^a=gi,e^b=egi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34151);