Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,32)(28,43)(29,38)(30,40)(33,41)(34,39)(35,42)(37,45)(44,47)(46,48);(2,6)(4,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,27,40)(20,28,33,46)(21,34,22,29)(25,37,42,47)(26,39,36,38)(31,43,41,48)(32,45,35,44);(1,3)(2,7)(4,6)(5,8)(11,16)(13,15)(17,20,22,32,23,33,21,35)(18,25,26,31,27,42,36,41)(19,28,29,45,30,46,34,44)(24,37,39,43,40,47,38,48);(2,6)(3,8)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,34)(20,32,33,35)(24,38,40,39)(25,41,42,31)(28,44,46,45)(37,43,47,48);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6877467);