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)(25,26,29,34,30,33,44,50)(27,35,39,46,37,31,38,47)(28,40,43,48,42,49,41,32)(36,52,54,45,53,56,55,51);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,37)(26,31,33,35)(28,36,42,53)(29,39,44,38)(32,45,48,51)(34,47,50,46)(40,56,49,52)(41,55,43,54);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,30,42)(26,32,33,48)(27,36,37,53)(29,41,44,43)(31,45,35,51)(34,49,50,40)(38,54,39,55)(46,56,47,52);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(26,33)(27,37)(32,48)(34,50)(36,53)(38,39)(40,49)(54,55);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,30,44)(26,34,33,50)(27,38,37,39)(28,41,42,43)(31,46,35,47)(32,49,48,40)(36,54,53,55)(45,56,51,52)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=f,d^2=h,e^2=hi,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, 6776222);