Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | |
1160 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)(4,8)(5,6)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,32)(28,33)(29,43)(30,35)(31,36)(34,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,51)(50,54)(52,55)(53,56);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,38)(26,32,36,46)(28,39,41,54)(29,30,44,40)(33,47,49,50)(34,48,43,35)(37,52,53,42)(45,55,56,51);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,40,52,31,41,30,42)(26,33,48,55,36,49,35,51)(27,37,44,54,38,53,29,39)(32,45,34,50,46,56,43,47);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(28,41)(33,49)(37,53)(39,54)(42,52)(45,56)(47,50)(51,55);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,30,38,31,44,40,27)(26,34,35,32,36,43,48,46)(28,39,42,53,41,54,52,37)(33,50,51,45,49,47,55,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fhi,d^2=fg,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6753917);