Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395552 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,29,15,19,14,18)(11,20,22,16,24,32,23,33)(12,25,30,17,28,35,27,36)(21,31,26,34,38,40,37,39);(2,5)(6,7)(9,11,13,22,15,24,14,23)(10,16,18,20,19,33,29,32)(12,26,27,21,28,37,30,38)(17,34,35,40,36,39,25,31);(9,12)(10,17)(11,21)(13,27)(14,30)(15,28)(16,31)(18,35)(19,36)(20,34)(22,37)(23,26)(24,38)(25,29)(32,39)(33,40);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=fi,b^2=eh,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323372);