Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,33,31,36,30,35)(27,37,39,48,41,46,40,32)(28,42,44,50,43,34,45,49)(38,52,54,47,53,51,55,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,41)(26,32,36,48)(28,38,43,53)(29,40,30,39)(33,46,35,37)(34,47,42,56)(44,55,45,54)(49,52,50,51);(11,17)(14,21)(18,19)(22,23)(26,33)(29,30)(32,46)(34,49)(35,36)(37,48)(39,40)(42,50)(44,45)(47,52)(51,56)(54,55);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28)(26,34)(27,38)(29,44)(30,45)(31,43)(32,47)(33,49)(35,50)(36,42)(37,51)(39,54)(40,55)(41,53)(46,52)(48,56);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(28,43)(34,42)(38,53)(44,45)(47,56)(49,50)(51,52)(54,55)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393607);