Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,32)(28,34)(29,35)(30,45)(31,36)(33,41)(37,47)(38,48)(39,50)(40,54)(42,52)(43,49)(44,53)(46,56)(51,55);(1,2,4,5)(3,7,8,6)(11,17)(12,20)(14,21)(15,19)(25,27,31,39)(26,32,36,50)(28,40,42,37)(29,43,30,38)(33,51,53,46)(34,54,52,47)(35,49,45,48)(41,55,44,56);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,31,42)(26,33,36,53)(27,37,39,40)(29,41,30,44)(32,46,50,51)(34,35,52,45)(38,55,43,56)(47,49,54,48);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(25,28)(26,34)(27,37)(29,44)(30,41)(31,42)(32,47)(33,45)(35,53)(36,52)(38,56)(39,40)(43,55)(46,48)(49,51)(50,54);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,31,30)(26,35,36,45)(27,38,39,43)(28,41,42,44)(32,48,50,49)(33,52,53,34)(37,55,40,56)(46,47,51,54)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,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, 7075791);