Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 | |
1245 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)(25,26,31,34,30,36,29,35)(27,32,39,51,38,47,40,48)(28,41,43,33,45,52,44,50)(37,53,55,46,42,56,54,49);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,38)(26,32,36,47)(28,42,45,37)(29,40,31,39)(33,49,50,46)(34,51,35,48)(41,56,52,53)(43,54,44,55);(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)(29,43)(30,45)(31,44)(32,46)(34,41)(35,52)(36,50)(38,42)(39,54)(40,55)(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)(37,42)(39,40)(46,49)(48,51)(53,56)(54,55);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,29,30,31)(26,34,36,35)(27,39,38,40)(28,43,45,44)(32,48,47,51)(33,41,50,52)(37,54,42,55)(46,56,49,53)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=f,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6775775);