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 | #17086 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,18,21,22)(19,23,24,20)(25,26,28,30)(27,31,32,29);(2,5)(3,8)(9,11,13,16)(10,14,15,12)(17,19,21,24)(18,20,22,23)(27,32)(29,31);(2,5)(6,7)(11,16)(12,14)(17,18,21,22)(19,23,24,20)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,29)(28,32)(30,31);(1,3,4,8)(2,6,5,7)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,e^2=fh,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, 1695998);