Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #239359 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1178 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 188 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,24,32)(12,25,27,38)(13,30,14,18)(16,23,33,22)(17,29,35,28)(21,31,36,39)(26,34,37,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,24)(10,16,19,33)(12,26,27,37)(13,23,14,22)(17,34,35,40)(18,32,30,20)(21,28,36,29)(25,31,38,39);(5,8)(9,12,13,28,15,27,14,29)(10,17,18,25,19,35,30,38)(11,21,22,37,24,36,23,26)(16,31,20,34,33,39,32,40);(12,27)(17,35)(21,36)(25,38)(26,37)(28,29)(31,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,d^c=dh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146557);