Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #38079 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1834 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)(17,18,22,26)(19,21,25,23)(20,27,30,28)(24,29,32,31);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,26)(22,25)(24,31)(27,30)(29,32);(10,13)(14,16)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,31,23,29);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,18)(19,21)(20,24)(22,26)(23,25)(27,31)(28,29)(30,32);(3,8)(5,7)(10,13)(14,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,32,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6675199);