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 | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,32)(28,39)(29,34)(30,44)(31,35)(33,41)(36,53)(37,47)(38,48)(40,56)(42,51)(43,49)(45,52)(46,55)(50,54);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,37)(26,32,35,47)(28,40,42,36)(29,43,30,38)(33,50,52,46)(34,49,44,48)(39,56,51,53)(41,54,45,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,36)(29,41)(30,45)(31,42)(32,46)(34,51)(35,52)(37,40)(38,54)(39,44)(43,55)(47,50)(48,53)(49,56);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(27,37)(32,47)(36,40)(38,43)(46,50)(48,49)(53,56)(54,55);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,31,30)(26,34,35,44)(27,38,37,43)(28,41,42,45)(32,48,47,49)(33,51,52,39)(36,54,40,55)(46,53,50,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,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, 6767631);