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 | #33179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1609 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(19,28)(23,37)(29,39)(31,33)(32,43)(38,46)(44,48)(45,47);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,18,22,26)(19,29,28,39)(20,24,30,27)(21,35,36,40)(23,31,37,33)(25,41,42,34)(32,46,43,38)(44,45,48,47);(9,12,13,11)(10,14,15,16)(17,19,30,31,22,28,20,33)(18,23,24,39,26,37,27,29)(21,32,41,45,36,43,34,47)(25,38,40,48,42,46,35,44);(2,5)(6,8)(17,20,22,30)(18,24,26,27)(19,31,28,33)(21,34,36,41)(23,29,37,39)(25,40,42,35)(32,45,43,47)(38,44,46,48);(1,3,4,7)(2,6,5,8)(17,21,30,41,22,36,20,34)(18,25,24,40,26,42,27,35)(19,32,31,45,28,43,33,47)(23,38,39,48,37,46,29,44)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,d^2=i,e^2=fhi,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, 6727824);