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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,31,36)(27,32,38,45)(28,33,40,47)(29,34,42,49)(30,35,43,50)(37,44,52,55)(39,46,53,56)(41,48,51,54);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,31,38)(26,32,36,45)(28,39,40,53)(29,30,42,43)(33,46,47,56)(34,35,49,50)(37,51,52,41)(44,54,55,48);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,30,41,31,40,43,51)(26,33,35,48,36,47,50,54)(27,37,29,39,38,52,42,53)(32,44,34,46,45,55,49,56);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,40)(33,47)(37,52)(39,53)(41,51)(44,55)(46,56)(48,54);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,29,30,38,31,42,43,27)(26,34,35,45,36,49,50,32)(28,39,41,52,40,53,51,37)(33,46,48,55,47,56,54,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=f,e^2=gh,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, 6738644);