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)(28,39,42,32)(30,34,43,48)(36,49,51,44)(37,45,41,47)(40,53,52,46)(50,55,54,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(27,36)(28,40)(30,43)(31,44)(32,46)(34,48)(35,49)(37,41)(38,51)(39,53)(42,52)(45,47);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,30,36,29,38,43,51)(26,31,34,44,33,35,48,49)(28,41,40,54,42,37,52,50)(32,47,46,56,39,45,53,55);(10,16)(14,17)(15,22)(21,23)(28,42)(32,39)(37,41)(40,52)(45,47)(46,53)(50,54)(55,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,28,30,40,29,42,43,52)(26,32,34,46,33,39,48,53)(27,37,36,50,38,41,51,54)(31,45,44,55,35,47,49,56)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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6743142);