Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
1213 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,35)(27,36,39,32)(28,40,43,33)(29,44,31,34)(37,47,55,53)(38,48,45,49)(41,50,54,56)(42,51,46,52);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,39)(26,32,35,36)(28,41,43,54)(29,38,31,45)(33,50,40,56)(34,48,44,49)(37,46,55,42)(47,51,53,52);(2,5)(6,8)(25,28,30,43)(26,33,35,40)(27,37,39,55)(29,42,31,46)(32,47,36,53)(34,51,44,52)(38,54,45,41)(48,50,49,56);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,29,30,31)(26,34,35,44)(27,38,39,45)(28,42,43,46)(32,48,36,49)(33,51,40,52)(37,54,55,41)(47,50,53,56);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,38)(28,43)(29,31)(32,49)(33,40)(34,44)(36,48)(37,41)(39,45)(47,50)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6611285);