Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190107 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1696 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 1696 conjugacy classes of subgroups — too many to draw. Only its 254 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 — 4 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,14,20,15,19,13,18)(11,21,24,31,25,16,23,32)(12,26,29,33,28,17,27,34)(22,30,38,40,37,35,36,39);(9,11,15,25)(10,16,19,21)(12,22,28,37)(13,24,14,23)(17,30,26,35)(18,32,20,31)(27,38,29,36)(33,40,34,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17,19,26)(11,22)(13,27)(14,29)(15,28)(16,30,21,35)(18,33,20,34)(23,36)(24,38)(25,37)(31,39,32,40);(9,13,15,14)(10,18,19,20)(11,23,25,24)(12,27,28,29)(16,31,21,32)(17,33,26,34)(22,36,37,38)(30,39,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97336);