Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | |
1153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,35,30,36,29,44)(27,37,40,49,41,32,45,50)(28,34,42,53,43,52,46,33)(38,48,56,47,39,54,55,51);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,41)(26,32,36,37)(28,39,43,38)(29,45,31,40)(33,51,53,47)(34,48,52,54)(35,50,44,49)(42,55,46,56);(1,2)(3,8)(4,7)(5,6)(25,28,30,43)(26,33,36,53)(27,38,41,39)(29,42,31,46)(32,47,37,51)(34,35,52,44)(40,55,45,56)(48,49,54,50);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,39)(29,46)(30,43)(31,42)(32,48)(33,44)(35,53)(36,52)(37,54)(38,41)(40,55)(45,56)(47,50)(49,51);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,29,30,31)(26,35,36,44)(27,40,41,45)(28,42,43,46)(32,49,37,50)(33,52,53,34)(38,55,39,56)(47,54,51,48)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,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, 7397495);