Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324564 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1303 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,34)(27,35,39,51)(28,32,41,48)(29,43,36,33)(31,44,46,38)(37,50,40,54)(42,56,52,49)(45,55,47,53);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(1,3)(2,5)(4,8)(6,7)(25,27,36,44,30,39,29,38)(26,31,43,51,34,46,33,35)(28,40,52,53,41,37,42,55)(32,47,56,50,48,45,49,54);(2,6)(3,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,30,41)(26,32,34,48)(27,37,39,40)(29,42,36,52)(31,45,46,47)(33,49,43,56)(35,50,51,54)(38,53,44,55);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(28,41)(32,48)(37,40)(42,52)(45,47)(49,56)(50,54)(53,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7236038);