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 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,26,30,34)(27,31,36,44)(28,33,39,48)(29,32,41,35)(37,43,49,55)(38,45,50,42)(40,46,52,47)(51,54,56,53);(1,2,5,6)(3,7,4,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,37)(29,38)(30,36)(32,45)(33,43)(34,44)(35,42)(39,49)(40,51)(41,50)(46,54)(47,53)(48,55)(52,56);(1,3,5,4)(2,7,6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(27,36)(28,39)(29,41)(31,42)(33,47)(34,35)(43,54)(44,45)(46,48)(51,56)(53,55);(2,6)(7,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,37)(29,40)(30,39)(31,43)(32,46)(34,48)(35,47)(36,49)(38,51)(41,52)(42,53)(44,55)(45,54)(50,56);(1,4,5,3)(2,8,6,7)(10,16)(14,17)(15,24)(22,23)(26,34)(28,39)(31,44)(32,35)(37,49)(40,52)(42,45)(51,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fhi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 7036677);