Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #30475 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1395 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)(4,8)(5,6)(9,10,13,15)(11,16,12,14)(17,18,22,27)(19,24,30,39)(20,32,35,25)(21,26,31,40)(23,28,34,42)(29,45,47,38)(33,43,36,41)(37,48,46,44);(9,10)(11,14)(12,16)(13,15)(17,19,22,30)(18,24,27,39)(20,33,35,36)(21,34,31,23)(25,41,32,43)(26,42,40,28)(29,46,47,37)(38,48,45,44);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,34,46,22,35,23,37)(18,25,42,48,27,32,28,44)(19,29,21,36,30,47,31,33)(24,38,26,43,39,45,40,41);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(20,35)(25,32)(29,47)(33,36)(37,46)(38,45)(41,43)(44,48);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,23,19,22,31,34,30)(18,26,28,24,27,40,42,39)(20,36,37,29,35,33,46,47)(25,43,44,38,32,41,48,45)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fhi,d^2=gi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6747536);