Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660402 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1439 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,31)(29,32);(1,3,6,5)(2,7,4,8)(10,13)(14,16)(17,19,22,21)(18,23,26,24)(20,28,29,27)(25,31,32,30);(1,4,6,2)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(18,24)(19,21)(20,29)(23,26)(25,30)(31,32);(2,4)(7,8)(10,13)(14,16)(17,20,22,29)(18,25,26,32)(19,27,21,28)(23,30,24,31);(1,5,6,3)(2,8,4,7)(9,11,12,15)(10,14,13,16)(20,29)(25,32)(27,28)(30,31)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574597);