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 | |
1369 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)(25,26,30,32)(27,36,37,48)(28,40,42,34)(29,35,31,45)(33,43,46,39)(38,53,52,51)(41,50,44,49)(47,55,56,54);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,32)(27,37)(34,40)(35,45)(38,52)(39,43)(49,50)(54,55);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,29,43,30,37,31,39)(26,33,45,48,32,46,35,36)(28,38,44,54,42,52,41,55)(34,47,49,53,40,56,50,51);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28,30,42)(26,34,32,40)(27,38,37,52)(29,44,31,41)(33,47,46,56)(35,50,45,49)(36,51,48,53)(39,55,43,54);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,30,31)(26,35,32,45)(27,39,37,43)(28,41,42,44)(33,48,46,36)(34,49,40,50)(38,54,52,55)(47,51,56,53)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=i,e^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=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820131);