Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(11,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26,29,30)(27,33,34,31)(28,32,36,38)(35,40,37,39);(2,3)(6,7)(17,19,21,24)(18,22,23,20)(26,30)(27,34)(28,36)(39,40);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,38)(37,39);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,34)(26,31,30,33)(28,37,36,35)(32,40,38,39);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,20)(18,19)(21,22)(23,24)(25,28,29,36)(26,32,30,38)(27,35,34,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fghi,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3238830);