Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(25,26,31,33,30,35,29,36)(27,32,43,48,37,47,39,45)(28,40,44,50,41,34,42,49)(38,52,54,55,51,46,53,56);(1,3,5,4)(2,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,30,37)(26,32,35,47)(28,38,41,51)(29,39,31,43)(33,48,36,45)(34,46,40,52)(42,53,44,54)(49,56,50,55);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,33)(27,37)(28,41)(29,31)(32,45)(34,50)(35,36)(40,49)(46,56)(47,48)(52,55)(53,54);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,38)(29,42)(30,41)(31,44)(32,46)(33,49)(35,40)(36,50)(37,51)(39,53)(43,54)(45,55)(47,52)(48,56);(1,4,5,3)(2,6,8,7)(10,16)(14,17)(15,24)(22,23)(26,35)(27,37)(28,41)(33,36)(39,43)(42,44)(46,52)(55,56)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=hi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,d^c=dh,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, 6439186);