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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,12)(11,14,15,16)(19,23)(21,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,35)(36,39)(38,40);(1,3,5,7)(2,4,6,8)(11,15)(14,16)(26,27)(28,35)(29,37)(38,39);(1,4)(2,3)(5,8)(6,7)(9,11,13,15)(10,14,12,16)(17,18,20,22)(19,21,23,24)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,38,40,39);(1,3)(2,4)(5,7)(6,8)(9,10,13,12)(11,14,15,16)(17,19,20,23)(18,21,22,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,2)(3,4)(5,6)(7,8)(9,12,13,10)(11,16,15,14)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,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, 1594608);