Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | |
1277 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,41)(29,34)(30,46)(31,35)(32,39)(33,43)(37,54)(38,48)(40,53)(42,52)(44,51)(45,50)(47,56)(49,55);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,38,31,40,30,39)(26,32,46,53,35,48,34,36)(28,42,45,55,43,37,44,56)(33,49,50,52,41,47,51,54);(2,6)(7,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,45)(30,44)(31,43)(32,47)(34,51)(35,41)(36,52)(38,56)(39,55)(40,42)(46,50)(48,49)(53,54);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(28,43)(33,41)(37,42)(44,45)(47,49)(50,51)(52,54)(55,56);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,46)(27,38,40,39)(28,44,43,45)(32,36,48,53)(33,50,41,51)(37,55,42,56)(47,54,49,52)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6472729);