Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
654 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 178 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:
(1,2,5,8)(3,6,7,4)(9,10,15,20)(11,21,25,40)(12,26,28,37)(13,17,30,18)(14,31,32,19)(16,33,35,24)(22,34,27,36)(23,38,29,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,24,15,25,32,33)(10,16,19,21,20,35,31,40)(12,27,13,29,28,22,30,23)(17,34,26,39,18,36,37,38);(1,4,5,6)(2,7,8,3)(9,12,15,28)(10,17,20,18)(11,22,25,27)(13,14,30,32)(16,34,35,36)(19,37,31,26)(21,38,40,39)(23,24,29,33);(2,8)(3,7)(9,13,14,28,15,30,32,12)(10,18,19,37,20,17,31,26)(11,23,24,27,25,29,33,22)(16,34,21,39,35,36,40,38)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384677);