Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1321 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)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,31,36)(27,32,40,50)(28,34,42,52)(29,35,30,45)(33,44,53,41)(37,54,38,47)(39,48,43,49)(46,55,51,56);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,31,40)(26,32,36,50)(28,38,42,37)(29,43,30,39)(33,51,53,46)(34,47,52,54)(35,49,45,48)(41,55,44,56);(3,8)(5,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,28,31,42)(26,33,36,53)(27,37,40,38)(29,41,30,44)(32,46,50,51)(34,35,52,45)(39,55,43,56)(47,48,54,49);(2,6)(5,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28)(26,34)(27,38)(29,44)(30,41)(31,42)(32,47)(33,45)(35,53)(36,52)(37,40)(39,55)(43,56)(46,49)(48,51)(50,54);(11,19)(15,17)(18,24)(22,23)(25,29,31,30)(26,35,36,45)(27,39,40,43)(28,41,42,44)(32,48,50,49)(33,52,53,34)(37,55,38,56)(46,54,51,47)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fh,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6861472);