Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ 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:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,34)(27,36,38,31)(28,33,41,50)(30,32,43,35)(37,48,40,45)(39,46,52,44)(42,47,54,49)(51,56,53,55);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,31)(28,40)(29,38)(30,39)(32,44)(33,48)(34,36)(35,46)(37,41)(42,53)(43,52)(45,50)(47,56)(49,55)(51,54);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(26,32)(28,41)(30,43)(31,44)(33,49)(34,35)(36,46)(37,40)(39,52)(45,56)(47,50)(48,55);(1,4,5,3)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28,29,41)(26,33,34,50)(27,37,38,40)(30,42,43,54)(31,45,36,48)(32,47,35,49)(39,51,52,53)(44,55,46,56);(10,16)(11,18)(15,24)(17,19)(27,38)(28,41)(31,36)(33,50)(39,52)(42,54)(44,46)(47,49)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6943595);