Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18)(19,28)(20,31)(21,26)(22,27)(23,24)(25,29)(30,41)(32,47)(33,44)(34,38)(35,48)(36,43)(37,40)(39,45)(42,46);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,29,33,37)(21,34,36,30)(25,31,40,44)(26,41,43,38)(32,46,35,45)(39,48,42,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,33)(18,25,27,40)(19,29,23,37)(21,35,36,32)(24,31,28,44)(26,42,43,39)(30,46,34,45)(38,48,41,47);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,22,36)(18,26,27,43)(19,30,23,34)(20,32,33,35)(24,38,28,41)(25,39,40,42)(29,45,37,46)(31,47,44,48);(2,5)(6,8)(10,14)(13,15)(18,27)(21,36)(24,28)(25,40)(30,34)(31,44)(32,35)(45,46)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7143821);