Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,32)(28,34)(29,35)(30,33)(31,36)(37,45)(38,46)(39,44)(40,47)(41,50)(42,48)(43,49)(51,56)(52,54)(53,55);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,40)(26,32,36,47)(28,37,41,51)(29,38,30,39)(33,44,35,46)(34,45,50,56)(42,52,43,53)(48,54,49,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(26,33)(28,41)(29,30)(32,44)(34,49)(35,36)(37,51)(38,39)(45,55)(46,47)(48,50)(54,56);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28,31,41)(26,34,36,50)(27,37,40,51)(29,43,30,42)(32,45,47,56)(33,48,35,49)(38,53,39,52)(44,54,46,55);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,35,36,33)(27,38,40,39)(28,42,41,43)(32,46,47,44)(34,48,50,49)(37,52,51,53)(45,54,56,55)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6947818);