Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| 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 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 850 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)(11,21,25,17)(12,26,24,36)(13,18,30,34)(14,19)(15,20)(16,23)(22,37)(27,35)(28,31)(29,33,38,40)(32,39);(9,11,15,25)(10,16,20,32)(12,22,24,14)(13,23,30,39)(17,18,21,34)(19,35,37,31)(26,40,36,33)(27,38,28,29);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,22)(13,28)(14,25)(15,24)(16,18)(19,36)(20,21)(23,29)(26,37)(27,30)(31,40)(32,34)(33,35)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=ei,f^b=fg,d^c=di,e^c=ei,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, 36530);