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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,11)(7,9,8,12)(13,14)(15,16)(17,19)(18,20)(21,22,26,27)(23,31,32,28)(24,34,35,29)(25,33,36,30);(5,7,10,8)(6,9,11,12)(13,15)(14,18)(16,19)(17,20)(22,27)(23,32)(24,35)(25,36);(6,11)(7,8)(13,14,17,19)(15,18,20,16)(21,23,26,32)(22,28,27,31)(24,33,35,30)(25,29,36,34);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,16)(14,15)(17,18)(19,20)(21,24,26,35)(22,29,27,34)(23,33,32,30)(25,28,36,31);(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,25,26,36)(22,30,27,33)(23,34,32,29)(24,28,35,31)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,d^2=i,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, 4990226);