Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #38033 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,4,8,5)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,25,21,24)(20,28,30,31)(23,29,32,27);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,24,26,25)(20,27,30,29)(23,28,32,31);(1,3)(2,4)(5,7)(6,8)(11,16)(13,15)(17,20,23,24,22,30,32,25)(18,19,27,28,26,21,29,31);(1,4,6,5)(2,3,7,8)(10,14)(13,15)(17,21,22,19)(18,25,26,24)(20,29,30,27)(23,31,32,28);(1,5,6,4)(2,3,7,8)(19,21)(20,30)(24,25)(28,31)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897256);