Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1600 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)(17,18,24,38,23,28,22,27)(19,29,21,25,31,45,32,41)(20,26,39,48,35,43,36,44)(30,46,37,40,33,47,34,42);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,25,28,41)(20,33,35,30)(21,24,32,22)(26,42,43,40)(27,45,38,29)(34,39,37,36)(44,46,48,47);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,30)(21,37)(22,36)(23,35)(24,39)(25,40)(27,44)(28,43)(29,46)(31,33)(32,34)(38,48)(41,42)(45,47);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,32,24,31)(18,25,27,29,28,41,38,45)(20,34,36,33,35,37,39,30)(26,42,44,47,43,40,48,46);(10,14)(13,15)(19,31)(20,35)(21,32)(25,41)(26,43)(29,45)(36,39)(44,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=hi,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6987831);