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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1339 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)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,35)(27,36,39,47)(28,32,40,49)(29,33,44,34)(31,45,48,38)(37,53,41,56)(42,51,43,52)(46,54,50,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(28,40)(32,49)(37,41)(42,43)(46,50)(51,52)(53,56)(54,55);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,38,30,39,44,45)(26,31,34,36,35,48,33,47)(28,41,43,54,40,37,42,55)(32,50,51,56,49,46,52,53);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28,30,40)(26,32,35,49)(27,37,39,41)(29,43,44,42)(31,46,48,50)(33,52,34,51)(36,53,47,56)(38,55,45,54);(3,7)(5,8)(11,21)(15,18)(19,20)(23,24)(25,29,30,44)(26,33,35,34)(27,38,39,45)(28,42,40,43)(31,47,48,36)(32,51,49,52)(37,54,41,55)(46,53,50,56)Polycyclic presentation:
2^9|a^2=fi,c^2=fgh,d^2=fi,e^2=i,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=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670939);