Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1350 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,34)(27,35,38,49)(28,32,40,46)(29,42,36,33)(31,43,45,39)(37,48,51,56)(41,53,50,47)(44,54,55,52);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(28,40)(32,46)(37,51)(41,50)(44,55)(47,53)(48,56)(52,54);(2,6)(3,7)(25,27,36,43,30,38,29,39)(26,31,42,49,34,45,33,35)(28,37,50,54,40,51,41,52)(32,44,53,56,46,55,47,48);(10,15)(14,17)(16,23)(22,24)(25,28)(26,32)(27,37)(29,41)(30,40)(31,44)(33,47)(34,46)(35,48)(36,50)(38,51)(39,52)(42,53)(43,54)(45,55)(49,56);(3,7)(5,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,38)(28,40)(31,45)(32,46)(35,49)(39,43)(41,50)(47,53)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6672180);