Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,31,36)(27,32,37,47)(28,34,41,50)(29,35,30,33)(38,45,51,55)(39,46,42,44)(40,48,43,49)(52,54,53,56);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,38)(29,42)(30,39)(31,37)(33,46)(34,45)(35,44)(36,47)(40,53)(41,51)(43,52)(48,56)(49,54)(50,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(27,37)(29,30)(32,44)(34,48)(35,36)(38,51)(40,43)(45,54)(46,47)(49,50)(55,56);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,34)(27,38)(29,43)(30,40)(31,41)(32,45)(33,48)(35,49)(36,50)(37,51)(39,53)(42,52)(44,54)(46,56)(47,55);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,31,30)(26,35,36,33)(27,39,37,42)(28,40,41,43)(32,46,47,44)(34,48,50,49)(38,52,51,53)(45,54,55,56)Polycyclic presentation:
2^9|a^2=fi,c^2=f,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6767479);