Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12721 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
644 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,21,25,37)(12,18,13,27)(14,19)(15,20)(16,23)(17,26,34,28)(22,33)(24,31,29,40)(30,35)(32,39)(36,38);(9,11)(10,16,20,33)(12,22)(13,23)(14,29)(15,25)(17,31,34,40)(18,21,27,37)(19,36,35,32)(24,30)(26,39)(28,38);(1,3,5,4)(2,6,8,7)(9,12)(10,17,20,34)(11,22)(13,15)(14,26)(16,31,33,40)(18,35,27,19)(21,32,37,36)(23,25)(24,38)(28,30)(29,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gh,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=ei,f^b=fgi,d^c=di,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36591);