Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130254 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #62328 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 623 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(2,5)(3,7)(6,8)(9,10,12,18,13,29,27,33,15,20,28,32,14,19,26,16)(11,21,24,39,30,38,40,35,25,37,22,31,23,17,34,36);(1,2)(3,8)(4,6)(5,7)(10,16)(12,26)(13,14)(17,31)(18,19)(20,33)(21,35)(22,40)(23,30)(24,34)(27,28)(29,32)(36,37)(38,39);(3,7)(5,8)(9,11)(10,17)(12,24)(13,30)(14,23)(15,25)(16,31)(18,36)(19,37)(20,38)(21,29)(22,28)(26,34)(27,40)(32,35)(33,39)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37635);