Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,36)(27,37,39,48)(28,42,43,50)(29,35,30,34)(32,41,47,40)(33,44,49,45)(38,46,52,56)(51,53,55,54);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,41,31,39,29,40)(26,32,35,48,36,47,34,37)(28,38,45,54,43,52,44,53)(33,46,42,55,49,56,50,51);(11,17)(14,21)(18,19)(22,23)(25,28,29,44,31,43,30,45)(26,33,34,50,36,49,35,42)(27,38,40,53,39,52,41,54)(32,46,37,51,47,56,48,55);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(28,43)(33,49)(38,52)(42,50)(44,45)(46,56)(51,55)(53,54);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(27,39)(32,47)(37,48)(38,52)(40,41)(46,56)(51,55)(53,54)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394565);