Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1179 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)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,30,35)(27,32,38,47)(28,40,42,33)(29,34,39,48)(31,36,44,51)(37,53,55,46)(41,52,45,49)(43,56,54,50);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,38)(26,32,35,47)(28,41,42,45)(29,44,39,31)(33,49,40,52)(34,51,48,36)(37,54,55,43)(46,56,53,50);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,31,43,30,42,44,54)(26,33,36,50,35,40,51,56)(27,37,39,41,38,55,29,45)(32,46,48,49,47,53,34,52);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(28,42)(33,40)(37,55)(41,45)(43,54)(46,53)(49,52)(50,56);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,31,27,30,39,44,38)(26,34,36,32,35,48,51,47)(28,41,43,55,42,45,54,37)(33,49,50,53,40,52,56,46)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fh,d^2=fgi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6753868);