Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1388 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)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,28)(19,29,32,45)(20,25,21,26)(22,36,37,27)(24,38,40,31)(30,44,34,47)(33,43,35,41)(39,48,42,46);(2,5)(6,7)(10,14)(11,12)(17,19,22,31,23,32,37,38)(18,24,27,29,28,40,36,45)(20,30,33,46,21,34,35,48)(25,39,41,44,26,42,43,47);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,23)(22,33)(24,39)(26,28)(27,41)(29,44)(31,46)(32,34)(35,37)(36,43)(38,48)(40,42)(45,47);(9,12,13,11)(10,15,14,16)(17,21,23,20)(18,26,28,25)(19,30,32,34)(22,35,37,33)(24,39,40,42)(27,43,36,41)(29,44,45,47)(31,46,38,48);(10,14)(15,16)(20,21)(25,26)(30,34)(33,35)(39,42)(41,43)(44,47)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,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, 6392160);