Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1635 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,26)(20,21)(22,25)(23,32)(24,28)(27,29)(30,31);(1,3,7,4)(2,5,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,24,25,26)(20,31,32,27)(21,29,23,30);(1,4,7,3)(2,5,8,6)(10,13)(14,15)(17,20,21,18,22,32,23,25)(19,27,29,26,28,31,30,24);(1,5,7,6)(2,3,8,4)(9,11,12,16)(10,14,13,15)(18,25)(19,28)(20,32)(29,30);(1,6,7,5)(2,3,8,4)(17,21,22,23)(18,20,25,32)(19,29,28,30)(24,31,26,27)Polycyclic presentation:
2^9|a^2=hi,b^2=fhi,c^2=fg,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893094);