Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 2097152 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,18)(15,16,19,20)(21,22,26,23)(24,29,32,27)(25,30,34,28)(31,35,33,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(22,23)(27,29)(28,30)(35,36);(2,4)(6,9)(10,11)(13,16,17,20)(14,15,18,19)(21,23,26,22)(24,29,32,27)(25,28,34,30)(31,35,33,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35);(5,7)(6,10)(8,12)(9,11)(13,14)(15,16)(17,18)(19,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4942310);