Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,24,29,22,39)(19,26,31,41,20,32,34,25)(21,27,37,43,38,44,36,48)(30,42,35,47,33,45,46,40);(9,10)(11,15)(12,14)(13,16)(17,19,24,20)(18,25,29,41)(21,35,38,46)(22,31,23,34)(26,28,32,39)(27,42,44,45)(30,36,33,37)(40,48,47,43);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,19)(18,26,29,32)(21,33,38,30)(22,31,23,34)(25,28,41,39)(27,42,44,45)(35,37,46,36)(40,43,47,48);(11,16)(13,15)(17,21,22,36,24,38,23,37)(18,27,39,48,29,44,28,43)(19,30,34,46,20,33,31,35)(25,40,32,45,41,47,26,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,28,29,39)(19,31,20,34)(21,36,38,37)(25,32,41,26)(27,43,44,48)(30,35,33,46)(40,45,47,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,e^2=g,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6998053);