Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,6,3)(4,8,5,7)(9,10,15,20)(11,21,25,32)(12,26,30,35)(13,19,14,18)(16,24,33,22)(17,28,36,29)(23,31,39,40)(27,37,38,34);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,27)(13,24)(14,22)(15,25)(17,34)(18,32)(19,21)(20,33)(23,28)(26,40)(29,39)(30,38)(31,35)(36,37);(1,4,6,5)(2,7,3,8)(11,22)(12,28)(13,14)(16,21)(17,35)(18,19)(24,25)(26,36)(27,38)(29,30)(32,33)(34,37);(1,5,6,4)(2,7,3,8)(9,12)(10,17)(11,23)(13,29)(14,28)(15,30)(16,31)(18,26)(19,35)(20,36)(21,37)(22,38)(24,27)(25,39)(32,34)(33,40)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,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^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385310);