Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52139 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,22,25,21,24,29,32)(19,26,23,31,27,20,28,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,20)(21,27)(22,23)(24,26)(25,30)(28,29)(31,32);(9,12,13,11)(10,14,15,16)(19,27)(20,26)(23,28)(30,31);(2,5)(6,8)(19,27)(20,26)(23,28)(30,31);(1,3,4,7)(2,6,5,8)(17,20,29,31,21,26,22,30)(18,23,32,19,24,28,25,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df,e^b=ei,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6312805);