Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1383 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)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,30)(20,34)(21,29)(22,27)(23,28)(24,32)(25,38)(26,36)(31,40)(33,45)(35,41)(37,42)(39,43)(44,47)(46,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,32)(18,24,28,30)(20,35,38,39)(21,22,36,33)(25,41,34,43)(26,27,29,45)(31,47,48,37)(40,42,46,44);(9,12,13,11)(10,14,15,16)(17,20,36,47,23,38,21,37)(18,25,26,42,28,34,29,44)(19,31,22,39,32,48,33,35)(24,40,45,43,30,46,27,41);(2,6)(5,7)(17,21,23,36)(18,26,28,29)(19,22,32,33)(20,37,38,47)(24,27,30,45)(25,42,34,44)(31,39,48,35)(40,41,46,43);(3,8)(5,7)(17,22,36,32,23,33,21,19)(18,27,26,24,28,45,29,30)(20,35,47,31,38,39,37,48)(25,43,42,46,34,41,44,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6727408);