Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
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)(25,26,30,35,31,36,29,34)(27,32,39,48,38,47,44,51)(28,40,42,50,43,52,45,33)(37,53,55,49,41,56,54,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,38)(26,32,36,47)(28,41,43,37)(29,44,30,39)(33,49,50,46)(34,51,35,48)(40,56,52,53)(42,54,45,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,37)(29,42)(30,45)(31,43)(32,46)(34,40)(35,52)(36,50)(38,41)(39,54)(44,55)(47,49)(48,56)(51,53);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(27,38)(32,47)(37,41)(39,44)(46,49)(48,51)(53,56)(54,55);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,31,30)(26,34,36,35)(27,39,38,44)(28,42,43,45)(32,48,47,51)(33,40,50,52)(37,54,41,55)(46,56,49,53)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6775695);