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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2635 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2391 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)(5,6,9,7)(8,10,12,11)(14,15)(16,20)(21,22)(23,29)(24,26)(25,32)(27,30)(28,33)(31,34)(35,36);(6,7)(8,12)(13,14,17,15)(16,19,20,18)(22,24)(23,30)(25,33)(34,35);(6,7)(10,11)(14,15)(18,19)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35);(5,8,9,12)(6,10,7,11)(13,16,17,20)(14,18,15,19)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,d^2=f,e^2=gh,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, 4378903);