Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1349 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,34)(27,35,38,31)(28,39,42,47)(29,43,45,33)(32,40,49,36)(37,46,54,51)(41,56,44,55)(48,52,50,53);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,34,35)(28,40,42,36)(29,37,45,54)(32,47,49,39)(33,46,43,51)(41,53,44,52)(48,56,50,55);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,36,30,42,38,40)(26,32,31,39,34,49,35,47)(29,44,37,53,45,41,54,52)(33,50,46,56,43,48,51,55);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,45)(26,33,34,43)(27,37,38,54)(28,41,42,44)(31,46,35,51)(32,48,49,50)(36,52,40,53)(39,55,47,56);(3,7)(5,8)(11,21)(15,18)(19,20)(23,24)(29,45)(33,43)(37,54)(41,44)(46,51)(48,50)(52,53)(55,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgh,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670868);