Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,33)(27,35,38,31)(28,39,41,32)(30,34,42,46)(36,47,50,43)(37,44,51,48)(40,52,53,45)(49,54,56,55);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(27,36)(28,40)(30,42)(31,43)(32,45)(34,46)(35,47)(37,51)(38,50)(39,52)(41,53)(44,48);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,30,36,29,38,42,50)(26,31,34,43,33,35,46,47)(28,37,40,49,41,51,53,56)(32,44,45,54,39,48,52,55);(10,16)(14,17)(15,22)(21,23)(28,41)(32,39)(37,51)(40,53)(44,48)(45,52)(49,56)(54,55);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,28,30,40,29,41,42,53)(26,32,34,45,33,39,46,52)(27,37,36,49,38,51,50,56)(31,44,43,54,35,48,47,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fghi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6743078);