Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1801 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,28,30,24)(20,33,35,47)(21,27,23,38)(25,36,42,37)(29,44,34,48)(31,43,32,40)(39,46,41,45);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,24,26,28)(20,34,35,29)(21,31,23,32)(25,41,42,39)(27,40,38,43)(33,44,47,48)(36,46,37,45);(10,14)(11,15)(17,20,21,36,22,35,23,37)(18,25,38,47,26,42,27,33)(19,29,31,45,30,34,32,46)(24,39,43,44,28,41,40,48);(1,3,4,7)(2,6,5,8)(18,26)(19,30)(20,35)(27,38)(31,32)(36,37)(39,41)(44,48);(2,5)(6,8)(11,15)(13,16)(17,21,22,23)(18,27,26,38)(19,31,30,32)(20,36,35,37)(24,40,28,43)(25,33,42,47)(29,45,34,46)(39,48,41,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670395);