Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111483 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1106 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,7,8,5)(4,6)(9,10)(11,20)(12,25,28,17)(13,29)(14,18)(15,19)(16,22)(21,33,38,37)(23,32)(24,36)(26,40,39,31)(27,34,30,35);(1,2)(3,5)(4,6)(7,8)(9,11,15,24)(10,16,19,32)(12,26,28,39)(13,23,14,22)(17,33,25,37)(18,36,29,20)(21,30,38,27)(31,34,40,35);(1,3)(2,5)(4,8)(6,7)(9,12,15,28)(10,17,19,25)(11,21,24,38)(13,27,14,30)(16,31,32,40)(18,34,29,35)(20,37,36,33)(22,26,23,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fh,g^c=ghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18681);