Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,7,4,6)(25,26,30,35)(27,31,38,46)(28,33,40,50)(29,32,42,34)(36,44,39,48)(37,43,52,45)(41,47,54,49)(51,55,53,56);(1,3,5,4)(2,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,30,38)(26,31,35,46)(28,39,40,36)(29,37,42,52)(32,43,34,45)(33,48,50,44)(41,53,54,51)(47,56,49,55);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(28,40)(29,42)(31,43)(33,49)(34,35)(36,39)(37,52)(44,56)(45,46)(47,50)(48,55);(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)(26,33)(27,36)(29,41)(30,40)(31,44)(32,47)(34,49)(35,50)(37,51)(38,39)(42,54)(43,55)(45,56)(46,48)(52,53);(1,4,5,3)(2,6,8,7)(10,16)(14,17)(15,24)(22,23)(28,40)(33,50)(36,39)(41,54)(44,48)(47,49)(51,53)(55,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=h,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,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, 6438134);