Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454974 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,8,4,6)(12,24)(16,30)(25,34)(31,38);(1,3,5,4)(2,6,7,8)(9,10,13,17,14,18,20,27)(11,19,22,29,23,32,26,15)(12,16,25,31,24,30,34,38)(21,33,36,39,35,40,37,28);(3,4)(6,8)(9,11,20,22,14,23,13,26)(10,15,27,19,18,29,17,32)(12,21,34,36,24,35,25,37)(16,28,38,33,30,39,31,40);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,21)(13,25)(14,24)(15,28)(17,31)(18,30)(19,33)(20,34)(22,36)(23,35)(26,37)(27,38)(29,39)(32,40)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=egh,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382917);