Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457102 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,16)(12,26)(13,15)(17,21)(19,20)(22,38)(23,24)(27,39)(28,29)(30,36)(31,40)(32,33);(1,3,5,8)(2,6,4,7)(9,10,13,16,14,19,15,20)(11,21,23,17,25,32,24,33)(12,18,28,30,29,35,26,36)(22,37,27,31,39,34,38,40);(2,4)(6,7)(9,11,15,23,14,25,13,24)(10,17,20,32,19,33,16,21)(12,27,26,39,29,38,28,22)(18,34,36,40,35,37,30,31);(1,4,5,2)(3,7,8,6)(9,12,14,29)(10,18,19,35)(11,22)(13,28,15,26)(16,30,20,36)(17,31)(21,37)(23,27)(24,38)(25,39)(32,34)(33,40)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370431);