Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1270 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)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,32)(27,36,39,48)(28,34,40,46)(29,35,31,44)(33,42,45,38)(37,51,52,56)(41,49,43,50)(47,53,55,54);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(28,40)(33,45)(35,44)(36,48)(37,52)(41,43)(53,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,31,38,30,39,29,42)(26,33,35,36,32,45,44,48)(28,37,41,54,40,52,43,53)(34,47,50,51,46,55,49,56);(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,37)(29,43)(30,40)(31,41)(32,46)(33,47)(35,50)(36,51)(38,54)(39,52)(42,53)(44,49)(45,55)(48,56);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(25,29,30,31)(26,35,32,44)(27,38,39,42)(28,41,40,43)(33,48,45,36)(34,49,46,50)(37,53,52,54)(47,51,55,56)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419752);