Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632048 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1336 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)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26,31,36)(27,32,38,47)(28,40,43,52)(29,35,30,34)(33,45,50,42)(37,53,41,56)(39,51,44,48)(46,54,49,55);(2,5)(3,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27)(26,32)(28,41)(29,44)(30,39)(31,38)(33,49)(34,51)(35,48)(36,47)(37,43)(40,56)(42,54)(45,55)(46,50)(52,53);(1,3,4,7)(2,6,5,8)(25,28,31,43)(26,33,36,50)(27,37,38,41)(29,42,30,45)(32,46,47,49)(34,40,35,52)(39,54,44,55)(48,56,51,53);(2,5)(6,8)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(27,38)(32,47)(37,41)(39,44)(46,49)(48,51)(53,56)(54,55);(10,16)(14,17)(15,20)(23,24)(25,29,31,30)(26,34,36,35)(27,39,38,44)(28,42,43,45)(32,48,47,51)(33,40,50,52)(37,54,41,55)(46,56,49,53)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=h,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6558827);