Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,35)(27,32,38,47)(28,39,41,49)(29,34,30,43)(33,44,50,40)(36,51,53,56)(37,46,42,48)(45,52,55,54);(3,7)(5,8)(10,14)(15,20)(16,21)(22,24)(25,27,31,38)(26,32,35,47)(28,36,41,53)(29,42,30,37)(33,45,50,55)(34,48,43,46)(39,51,49,56)(40,54,44,52);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,31,41)(26,33,35,50)(27,36,38,53)(29,40,30,44)(32,45,47,55)(34,49,43,39)(37,52,42,54)(46,51,48,56);(2,6)(3,7)(11,17)(15,20)(18,23)(22,24)(25,29,31,30)(26,34,35,43)(27,37,38,42)(28,40,41,44)(32,46,47,48)(33,49,50,39)(36,52,53,54)(45,51,55,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(27,38)(32,47)(36,53)(37,42)(45,55)(46,48)(51,56)(52,54)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=dfh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538607);