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 | #29721 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,30)(20,34)(21,36)(22,28)(23,29)(24,27)(25,33)(26,31)(32,44)(35,47)(37,48)(38,43)(39,42)(40,46)(41,45);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19,23,33)(18,25,29,30)(20,21,31,24)(22,35,38,45)(26,36,34,27)(28,41,43,47)(32,39,46,37)(40,42,44,48);(9,12,13,11)(10,14,15,16)(17,20,24,19,23,31,21,33)(18,26,27,25,29,34,36,30)(22,35,39,32,38,45,37,46)(28,41,42,40,43,47,48,44);(2,5)(6,8)(17,21,23,24)(18,27,29,36)(19,31,33,20)(22,37,38,39)(25,26,30,34)(28,42,43,48)(32,45,46,35)(40,41,44,47);(1,3,4,7)(2,6,5,8)(17,22,21,37,23,38,24,39)(18,28,36,48,29,43,27,42)(19,32,20,35,33,46,31,45)(25,40,26,41,30,44,34,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=i,e^2=fh,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, 6727818);