Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #40175 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1583 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,26,29,23)(20,24,30,25)(27,28,32,31);(1,3,4,8)(2,5,6,7)(17,19,21,29)(18,23,22,26)(20,28,30,31)(24,27,25,32);(10,14)(15,16)(17,20,21,30)(18,24,22,25)(19,27,29,32)(23,31,26,28);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,28)(21,30)(22,24)(23,32)(26,27)(29,31);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,22)(19,23)(20,30)(26,29)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6637652);