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 | #23871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1613 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)(19,30,32,42,20,26,34,25)(21,35,37,45,38,27,40,46)(31,43,36,41,33,44,47,48);(9,10)(11,15)(12,14)(13,16)(17,19,24,20)(18,25,29,42)(21,36,38,47)(22,32,23,34)(26,39,30,28)(27,44,35,43)(31,40,33,37)(41,45,48,46);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,19)(18,26,29,30)(21,33,38,31)(22,32,23,34)(25,39,42,28)(27,43,35,44)(36,37,47,40)(41,45,48,46);(11,16)(13,15)(17,21,22,40,24,38,23,37)(18,27,39,45,29,35,28,46)(19,31,34,47,20,33,32,36)(25,41,26,43,42,48,30,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,28,29,39)(19,32,20,34)(21,37,38,40)(25,26,42,30)(27,45,35,46)(31,47,33,36)(41,44,48,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=ghi,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6998346);