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) = C26×C4 |
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 | |
1921 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)(6,7)(10,11)(14,18)(15,20)(21,22)(23,29)(24,30)(25,33)(26,27)(28,35)(31,32)(34,36);(5,6,9,7)(8,10,12,11)(22,27)(23,30)(28,33)(31,36);(13,14,17,18)(15,19,20,16)(21,23)(22,24)(25,31)(26,30)(27,29)(28,32)(33,34)(35,36);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,34)(26,29)(27,30)(28,36)(31,33)(32,35);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,15,18,20)(21,25,26,35)(22,28,27,33)(23,31,30,36)(24,32,29,34)Polycyclic presentation:
2^9|b^2=fh,c^2=gh,d^2=gi,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, 4342040);