Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1813 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,23,24)(20,31,34,39)(21,35,38,26)(25,29,42,32)(30,40,36,43)(33,48,37,47)(41,46,44,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,36,38,30)(25,31,42,39)(26,43,35,40)(33,45,37,46)(41,48,44,47);(10,14)(11,15)(17,20,19,29,22,34,23,32)(18,25,24,39,27,42,28,31)(21,37,30,46,38,33,36,45)(26,44,40,47,35,41,43,48);(2,6)(5,7)(17,21)(18,26)(19,30)(20,33)(22,38)(23,36)(24,40)(25,41)(27,35)(28,43)(29,45)(31,47)(32,46)(34,37)(39,48)(42,44);(3,8)(5,7)(11,15)(13,16)(21,38)(26,35)(30,36)(33,37)(40,43)(41,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,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=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6669304);