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 | |
1230 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)(25,26,29,35,30,34,31,36)(27,37,40,49,39,32,41,48)(28,42,46,33,45,52,44,51)(38,53,55,50,43,56,54,47);(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)(26,32)(28,43)(29,40)(30,39)(31,41)(33,50)(34,37)(35,48)(36,49)(38,45)(42,53)(44,55)(46,54)(47,51)(52,56);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,38)(29,44)(30,45)(31,46)(32,47)(34,51)(35,42)(36,52)(37,50)(39,43)(40,54)(41,55)(48,56)(49,53);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(26,34)(27,39)(33,51)(35,36)(38,43)(40,41)(42,52)(54,55);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,30,31)(26,35,34,36)(27,40,39,41)(28,44,45,46)(32,48,37,49)(33,42,51,52)(38,54,43,55)(47,56,50,53)Polycyclic presentation:
2^9|a^2=fg,c^2=fi,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, 6776203);