Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,13,12)(11,15,16,14)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(1,3)(2,6)(4,7)(5,8)(18,22)(23,24)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(2,7)(3,8)(9,10,13,12)(11,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11,13,16)(10,14,12,15)(17,19,21,20)(18,23,22,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(1,4)(2,3)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,24,22,23)(25,28,29,37)(26,32,33,40)(27,34,35,36)(30,38,31,39)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fi,d^2=hi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1691848);