Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1638 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,26,36,41)(19,28,30,38,31,42,21,24)(20,32,35,48,34,25,39,40)(29,44,46,37,45,43,33,47);(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,26,38)(20,29,34,45)(21,36,30,22)(25,37,32,47)(27,42,41,28)(33,39,46,35)(40,44,48,43);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,25,26,32)(19,29,31,45)(21,33,30,46)(22,35,36,39)(24,37,38,47)(27,40,41,48)(28,43,42,44);(11,16)(13,15)(17,21,22,19,23,30,36,31)(18,24,27,28,26,38,41,42)(20,33,35,29,34,46,39,45)(25,37,40,43,32,47,48,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(20,34)(24,38)(27,41)(28,42)(29,45)(33,46)(35,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6998052);