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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,30,35)(27,36,31,32)(28,39,42,46)(29,43,44,34)(33,37,49,40)(38,47,45,50)(41,54,55,52)(48,53,56,51);(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,20)(18,23)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,38,44,45)(33,39,49,46)(34,47,43,50)(41,53,55,51)(48,52,56,54);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,27,37,30,42,31,40)(26,33,32,46,35,49,36,39)(29,41,38,51,44,55,45,53)(34,48,47,54,43,56,50,52);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(25,29)(26,34)(27,38)(28,41)(30,44)(31,45)(32,47)(33,48)(35,43)(36,50)(37,51)(39,52)(40,53)(42,55)(46,54)(49,56);(2,6)(5,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(28,42)(29,44)(33,49)(34,43)(37,40)(38,45)(39,46)(47,50)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,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^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819106);