Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,16)(12,15)(13,14)(17,18)(19,23)(20,24)(21,22)(25,26,30,32)(27,31,29,28);(1,3,6,8)(2,7,4,5)(10,14)(15,16)(17,18,21,22)(19,23,24,20)(25,27,30,29)(26,28,32,31);(9,11,13,12)(10,15,14,16)(17,19)(18,20)(21,24)(22,23)(25,28,30,31)(26,27,32,29);(1,4,6,2)(3,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,20)(21,22)(23,24)(25,29,30,27)(26,31,32,28);(1,5)(2,3)(4,8)(6,7)(9,12,13,11)(10,16,14,15)(17,20)(18,19)(21,23)(22,24)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=ghi,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 771624);