Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453974 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35150 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1512 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 202 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:
(1,2,5,4)(3,7,8,6)(9,10,13,17)(11,19,21,15)(12,23,24,35)(14,27,26,18)(16,29,33,25)(20,36,37,40)(22,31,28,34)(30,38,32,39);(1,3,5,8)(2,6,4,7)(9,11,13,21)(10,15,17,19)(12,20,24,37)(14,28,26,22)(16,32,33,30)(18,34,27,31)(23,40,35,36)(25,39,29,38);(2,4)(6,7)(9,12)(10,16)(11,20)(13,24)(14,29)(15,30)(17,33)(18,35)(19,32)(21,37)(22,39)(23,27)(25,26)(28,38)(31,36)(34,40);(1,4,5,2)(3,7,8,6)(12,24)(14,26)(16,33)(18,27)(20,37)(22,28)(30,32)(31,34)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361657);