Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,35,30,37,29,36)(27,32,39,54,41,50,40,49)(28,42,44,33,46,34,45,52)(38,53,56,47,43,48,55,51);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,41)(26,32,37,50)(28,43,46,38)(29,40,31,39)(33,51,52,47)(34,53,42,48)(35,54,36,49)(44,55,45,56);(1,2)(3,8)(4,7)(5,6)(25,28,30,46)(26,33,37,52)(27,38,41,43)(29,44,31,45)(32,47,50,51)(34,35,42,36)(39,55,40,56)(48,54,53,49);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,38)(29,45)(30,46)(31,44)(32,48)(33,36)(35,52)(37,42)(39,56)(40,55)(41,43)(47,49)(50,53)(51,54);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,29,30,31)(26,35,37,36)(27,39,41,40)(28,44,46,45)(32,49,50,54)(33,42,52,34)(38,55,43,56)(47,48,51,53)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7397905);