Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,4,8)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,34)(21,29)(22,27)(23,28)(24,32)(25,33)(26,35)(31,44)(36,48)(37,43)(38,42)(39,46)(40,47)(41,45);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,21)(22,36)(23,32)(25,29)(26,34)(27,40)(28,30)(31,45)(33,35)(37,46)(38,47)(39,41)(42,48)(43,44);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,20,35,32,23,33,21,19)(18,25,26,30,28,34,29,24)(22,36,45,46,38,47,37,31)(27,40,41,44,42,48,43,39);(10,14)(13,15)(17,21,23,35)(18,26,28,29)(19,20,32,33)(22,37,38,45)(24,34,30,25)(27,41,42,43)(31,36,46,47)(39,48,44,40);(17,22,21,37,23,38,35,45)(18,27,29,43,28,42,26,41)(19,31,33,47,32,46,20,36)(24,39,34,48,30,44,25,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gh,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727978);