Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2143 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)(6,9)(7,11)(15,19)(18,20)(21,22,26,25)(23,27,31,30)(24,32,34,28)(29,36,33,35);(6,9)(10,12)(14,17)(18,20)(22,25)(27,30)(28,32)(35,36);(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,23,26,31)(22,27,25,30)(24,33,34,29)(28,36,32,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(21,24)(22,28)(23,29)(25,32)(26,34)(27,35)(30,36)(31,33);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,22)(23,30,31,27)(24,32,34,28)(29,36,33,35)Polycyclic presentation:
2^9|a^2=f,c^2=fgh,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4407369);