Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1424 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)(11,15)(12,16)(13,14)(17,18)(19,29)(20,34)(21,26)(22,39)(23,28)(24,32)(25,37)(27,40)(30,43)(31,42)(33,45)(35,41)(36,47)(38,48)(44,46);(2,5)(6,8)(10,14)(11,12)(17,19,22,32,23,33,40,35)(18,24,27,29,28,41,39,45)(20,30,37,46,36,21,31,38)(25,26,34,44,42,43,47,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,37,23,36,40,31)(18,25,27,34,28,42,39,47)(19,30,32,46,33,21,35,38)(24,26,29,44,41,43,45,48);(9,12,13,11)(10,16,14,15)(17,21,23,30)(18,26,28,43)(19,31,33,37)(20,35,36,32)(22,38,40,46)(24,34,41,47)(25,29,42,45)(27,44,39,48);(10,14)(15,16)(20,36)(21,30)(25,42)(26,43)(31,37)(34,47)(38,46)(44,48)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390218);