Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,35)(28,39)(29,31)(30,43)(32,38)(33,40)(34,42)(36,48)(37,51)(41,46)(44,50)(45,56)(47,49)(52,53)(54,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,31)(27,36)(33,39)(34,43)(37,49)(38,50)(46,54)(52,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,30,38,29,36,42,50)(26,32,34,35,31,44,43,48)(28,37,41,52,40,49,55,56)(33,45,46,47,39,53,54,51);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28,29,40)(26,33,31,39)(27,37,36,49)(30,41,42,55)(32,45,44,53)(34,46,43,54)(35,47,48,51)(38,52,50,56);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,40)(33,39)(37,49)(41,55)(45,53)(46,54)(47,51)(52,56)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820048);