Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #117631 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1418 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25,29,34)(13,30)(14,18)(15,19)(16,22)(17,28,33,27)(21,31,26,40)(23,32)(24,36)(35,38,39,37);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,15,24,14,23)(10,16,18,20,19,32,30,36)(12,26,27,38,29,21,28,37)(17,31,25,39,33,40,34,35);(5,8)(9,12)(10,17)(11,21)(13,28)(14,27)(15,29)(16,31)(18,34)(19,33)(20,35)(22,38)(23,37)(24,26)(25,30)(32,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=eh,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 24827);