Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1233 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)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,39)(37,47)(38,52)(40,53)(41,48)(42,56)(43,50)(44,51)(46,55)(49,54);(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,39,31,40,28,37)(26,32,33,36,35,47,45,53)(29,42,43,54,44,38,41,55)(34,49,48,56,51,46,50,52);(2,5)(6,8)(11,17)(14,21)(18,19)(22,23)(25,28,31,30)(26,33,35,45)(27,37,40,39)(29,43,44,41)(32,36,47,53)(34,50,51,48)(38,55,42,54)(46,56,49,52);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29)(26,34)(27,38)(28,41)(30,43)(31,44)(32,46)(33,48)(35,51)(36,52)(37,54)(39,55)(40,42)(45,50)(47,49)(53,56);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(29,44)(34,51)(38,42)(41,43)(46,49)(48,50)(52,56)(54,55)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393328);