Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1916 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)(5,6,8,11)(7,9,12,10)(13,14)(15,19)(16,17)(18,20)(23,29)(24,31)(27,32)(28,33);(5,7)(6,9)(8,12)(10,11)(13,15,17,20)(14,18,16,19)(22,26)(27,32)(28,33)(34,36);(5,7,8,12)(6,10,11,9)(13,14,17,16)(15,18,20,19)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,18,20,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=g,d^2=hi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4991497);