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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,31,36)(27,32,41,49)(28,34,43,50)(29,35,30,45)(33,44,51,42)(37,52,38,47)(39,48,40,55)(46,54,56,53);(3,4)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,37)(29,39)(30,40)(31,41)(33,46)(34,52)(35,48)(36,49)(38,43)(42,53)(44,54)(45,55)(47,50)(51,56);(1,3,5,4)(2,6,8,7)(25,28)(26,33)(27,37)(29,42)(30,44)(31,43)(32,46)(34,45)(35,50)(36,51)(38,41)(39,53)(40,54)(47,48)(49,56)(52,55);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,34)(27,38)(29,44)(30,42)(31,43)(32,47)(33,45)(35,51)(36,50)(37,41)(39,53)(40,54)(46,48)(49,52)(55,56);(1,4,5,3)(2,7,8,6)(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,35,36,45)(27,39,41,40)(28,42,43,44)(32,48,49,55)(33,50,51,34)(37,53,38,54)(46,47,56,52)Polycyclic presentation:
2^9|a^2=fhi,c^2=h,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6755138);