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 | |
1217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,36)(27,32,40,47)(28,34,42,50)(29,35,30,33)(37,45,52,56)(38,46,39,44)(41,48,43,49)(51,54,53,55);(1,2,5,6)(3,8,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,40)(26,32,36,47)(28,37,42,52)(29,38,30,39)(33,44,35,46)(34,45,50,56)(41,51,43,53)(48,54,49,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(26,33)(29,30)(32,44)(34,48)(35,36)(38,39)(41,43)(45,54)(46,47)(49,50)(51,53)(55,56);(1,3,5,4)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,37)(29,43)(30,41)(31,42)(32,45)(33,48)(35,49)(36,50)(38,53)(39,51)(40,52)(44,54)(46,55)(47,56);(1,4,5,3)(2,7,6,8)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,35,36,33)(27,38,40,39)(28,41,42,43)(32,46,47,44)(34,48,50,49)(37,51,52,53)(45,54,56,55)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=f,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=dh,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, 6947064);