Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,30,33)(27,34,37,39)(28,29,32,35)(31,38,40,36);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,35,38,30,37,29,36)(26,31,32,39,33,40,28,34);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,22)(18,23)(25,28,29,33,30,32,35,26)(27,34,36,40,37,39,38,31);(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24)(25,29,30,35)(26,32,33,28)(27,36,37,38)(31,39,40,34);(12,19)(16,22)(18,23)(20,24)(27,37)(31,40)(34,39)(36,38)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6475945);