Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(10,12)(14,16)(17,18,21,22)(19,23,24,20)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(3,4)(6,7)(11,15)(14,16)(19,24)(20,23)(26,30)(28,36)(31,33)(34,40);(1,3,5,4)(2,6,8,7)(9,10,13,12)(11,14,15,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,4,5,3)(2,7,8,6)(9,11,13,15)(10,14,12,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,18)(19,20)(21,22)(23,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fh,c^2=fghi,d^2=fghi,e^2=gh,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, 1594591);