Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1818 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)(5,6,8,9)(7,11,12,10)(13,14,16,19)(15,17,20,18)(22,27)(23,29)(25,33)(32,36);(7,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,22,26,27)(23,30,29,24)(25,34,33,28)(31,32,35,36);(6,9)(10,11)(13,15,16,20)(14,18,19,17)(21,23,26,29)(22,24,27,30)(25,35,33,31)(28,36,34,32);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,36)(26,30)(27,29)(28,35)(31,34)(32,33);(5,7,8,12)(6,10,9,11)(13,14)(15,18)(16,19)(17,20)(21,25,26,33)(22,28,27,34)(23,31,29,35)(24,32,30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4057964);