Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(12,20)(15,21)(19,24)(22,23)(25,26,31,35)(27,32,30,45)(28,38,41,54)(29,34,43,50)(33,36,48,40)(37,46,44,56)(39,53,42,55)(47,51,49,52);(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,30)(26,32,35,45)(28,36,41,40)(29,37,43,44)(33,38,48,54)(34,46,50,56)(39,51,42,52)(47,53,49,55);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,30,40,31,41,27,36)(26,33,32,38,35,48,45,54)(29,42,44,51,43,39,37,52)(34,49,46,55,50,47,56,53);(1,4,5,3)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,34)(27,37)(28,39)(30,44)(31,43)(32,46)(33,47)(35,50)(36,51)(38,53)(40,52)(41,42)(45,56)(48,49)(54,55);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(29,43)(34,50)(37,44)(39,42)(46,56)(47,49)(51,52)(53,55)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7094144);