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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,33)(21,26)(22,36)(23,28)(25,34)(27,37)(29,43)(30,39)(31,40)(32,47)(35,42)(38,44)(41,46)(45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,38)(26,39)(27,41)(28,40)(33,43)(34,44)(35,45)(36,47)(37,46)(42,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,25,28,42)(19,29,31,30)(22,35,37,34)(24,38,40,48)(26,36,33,27)(32,45,46,44)(39,47,43,41);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,20)(18,26,28,33)(19,30,31,29)(22,35,37,34)(24,39,40,43)(25,36,42,27)(32,45,46,44)(38,47,48,41);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(19,31)(20,21)(24,40)(25,42)(26,33)(32,46)(34,35)(41,47)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6908433);