Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632082 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,35,30,34,42,32)(27,31,38,45,37,46,53,43)(28,39,41,47,40,33,48,49)(36,50,51,54,52,44,55,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,37)(26,31,34,46)(28,36,40,52)(29,38,42,53)(32,43,35,45)(33,44,39,50)(41,51,48,55)(47,54,49,56);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(29,42)(31,43)(33,47)(34,35)(38,53)(39,49)(41,48)(44,54)(45,46)(50,56)(51,55);(2,5)(6,7)(11,18)(14,22)(17,19)(21,23)(25,28)(26,33)(27,36)(29,41)(30,40)(31,44)(32,47)(34,39)(35,49)(37,52)(38,51)(42,48)(43,54)(45,56)(46,50)(53,55);(1,3,4,8)(2,6,5,7)(10,16)(14,22)(15,24)(21,23)(26,34)(27,37)(28,40)(32,35)(38,53)(41,48)(44,50)(54,56)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efhi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6551656);