Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(14,18)(15,20)(21,22)(23,27)(24,31)(25,33)(26,29)(28,34)(30,32)(35,36);(5,7)(6,11)(8,9)(10,12)(21,23,26,31)(22,27,29,24)(25,30,34,36)(28,35,33,32);(5,8)(6,7)(9,12)(10,11)(13,14,17,18)(15,19,20,16)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,35)(34,36);(1,2)(3,4)(5,9,10,6)(7,8,12,11)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,32)(26,27)(28,30)(29,31)(33,36)(34,35);(5,8)(6,12)(7,9)(10,11)(13,16,17,19)(14,15,18,20)(21,25,26,34)(22,28,29,33)(23,30,31,36)(24,32,27,35)Polycyclic presentation:
2^9|b^2=fh,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4326188);