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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(10,13)(11,15)(17,18,21,27,22,28,23,29)(19,30,37,41,33,24,32,46)(20,26,38,25,36,45,35,43)(31,44,47,42,34,40,48,39);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(17,19,22,33)(18,24,28,30)(20,34,36,31)(21,37,23,32)(25,42,43,39)(26,44,45,40)(27,46,29,41)(35,47,38,48);(9,10)(11,14)(12,13)(15,16)(17,20,22,36)(18,25,28,43)(19,31,33,34)(21,35,23,38)(24,39,30,42)(26,29,45,27)(32,47,37,48)(40,41,44,46);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,20)(18,26)(19,31)(21,38)(22,36)(23,35)(24,40)(25,27)(28,45)(29,43)(30,44)(32,48)(33,34)(37,47)(39,46)(41,42);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,28,29)(19,32,33,37)(20,35,36,38)(24,41,30,46)(25,26,43,45)(31,47,34,48)(39,44,42,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7079237);