Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,34,31,36,30,35)(27,37,39,32,41,47,40,48)(28,42,44,50,45,49,43,33)(38,52,54,56,55,46,53,51);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,31,41)(26,32,36,48)(28,38,45,55)(29,40,30,39)(33,46,50,52)(34,37,35,47)(42,56,49,51)(43,54,44,53);(1,4,5,3)(2,6,7,8)(10,15)(11,18)(16,22)(19,24)(25,28,31,45)(26,33,36,50)(27,38,41,55)(29,43,30,44)(32,46,48,52)(34,49,35,42)(37,51,47,56)(39,53,40,54);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,31,30)(26,34,36,35)(27,39,41,40)(28,43,45,44)(32,47,48,37)(33,49,50,42)(38,53,55,54)(46,56,52,51);(12,20)(16,22)(19,24)(21,23)(25,29,31,30)(26,34,36,35)(27,40,41,39)(28,43,45,44)(32,37,48,47)(33,49,50,42)(38,54,55,53)(46,51,52,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgi,c^2=fgi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6478004);