Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,4)(17,18,22,27)(19,23,29,26)(20,30,32,28)(21,25,24,31);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,23,27,26)(20,31,32,25)(21,28,24,30);(1,3)(2,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,29,31,22,32,19,25)(18,24,23,28,27,21,26,30);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,21)(18,25)(19,28)(20,23)(22,24)(26,32)(27,31)(29,30);(2,6)(7,8)(10,14)(13,15)(17,19,22,29)(18,23,27,26)(20,25,32,31)(21,30,24,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818626);