Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1409 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)(17,18,23,28)(19,29,31,41)(20,34,38,43)(21,27,22,26)(24,33,40,32)(25,37,44,36)(30,39,35,42)(45,47,48,46);(2,5)(6,8)(10,14)(11,12)(17,19,22,33,23,31,21,32)(18,24,27,41,28,40,26,29)(20,35,36,47,38,30,37,46)(25,42,34,45,44,39,43,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,21,37,23,38,22,36)(18,25,26,43,28,44,27,34)(19,30,32,47,31,35,33,46)(24,39,29,45,40,42,41,48);(9,12,13,11)(10,16,14,15)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47);(10,14)(15,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,36,38,37)(24,29,40,41)(25,34,44,43)(30,46,35,47)(39,48,42,45)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=hi,e^2=i,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390255);