Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499804 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(25,26)(27,32)(28,40)(29,35)(30,43)(31,34)(33,41)(36,50)(37,47)(38,46)(39,53)(42,49)(44,48)(45,52)(51,55)(54,56);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,31,37)(26,32,34,47)(28,36,42,51)(29,38,30,39)(33,45,48,56)(35,46,43,53)(40,50,49,55)(41,52,44,54);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28)(26,33)(27,36)(29,41)(30,44)(31,42)(32,45)(34,48)(35,49)(37,51)(38,52)(39,54)(40,43)(46,55)(47,56)(50,53);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(26,34)(27,37)(33,48)(35,43)(36,51)(38,39)(40,49)(52,54);(10,16)(14,17)(15,24)(22,23)(25,29,31,30)(26,35,34,43)(27,38,37,39)(28,41,42,44)(32,46,47,53)(33,49,48,40)(36,52,51,54)(45,55,56,50)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437207);