Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1271 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)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,24,31,39)(20,25,30,38)(21,26,32,40)(22,27,36,44)(29,37,46,48)(33,41,35,43)(34,42,45,47);(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,28,39)(20,32,30,21)(22,35,36,33)(25,40,38,26)(27,43,44,41)(29,45,46,34)(37,47,48,42);(2,6)(7,8)(11,16)(13,15)(17,20,21,19,23,30,32,31)(18,25,26,24,28,38,40,39)(22,35,34,46,36,33,45,29)(27,43,42,48,44,41,47,37);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,32)(18,26,28,40)(19,20,31,30)(22,34,36,45)(24,25,39,38)(27,42,44,47)(29,33,46,35)(37,41,48,43);(1,4,5,3)(2,8,6,7)(17,22,21,34,23,36,32,45)(18,27,26,42,28,44,40,47)(19,29,30,35,31,46,20,33)(24,37,38,43,39,48,25,41)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fh,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751371);