Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,35)(27,36,31,32)(28,39,42,47)(29,43,45,34)(33,37,50,40)(38,48,46,52)(41,56,44,55)(49,54,51,53);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,38,45,46)(33,39,50,47)(34,48,43,52)(41,54,44,53)(49,55,51,56);(2,6)(7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,37,30,42,31,40)(26,33,32,47,35,50,36,39)(29,44,38,54,45,41,46,53)(34,51,48,55,43,49,52,56);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,45)(26,34,35,43)(27,38,31,46)(28,41,42,44)(32,48,36,52)(33,49,50,51)(37,53,40,54)(39,55,47,56);(1,4,5,3)(2,7,6,8)(11,21)(15,18)(19,20)(23,24)(28,42)(29,45)(33,50)(34,43)(37,40)(38,46)(39,47)(48,52)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6787056);