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 | |
1672 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22,26,30)(23,31,32,27)(24,34,35,28)(25,33,36,29);(7,11)(9,12)(13,14,16,18)(15,19,20,17)(23,32)(25,36)(27,31)(29,33);(5,6,8,10)(7,9,11,12)(13,15,16,20)(14,17,18,19)(21,23,26,32)(22,27,30,31)(24,29,35,33)(25,34,36,28);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,35)(22,28,30,34)(23,33,32,29)(25,27,36,31);(5,7)(6,9)(8,11)(10,12)(13,15)(14,17)(16,20)(18,19)(21,25,26,36)(22,29,30,33)(23,34,32,28)(24,27,35,31)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=g,d^2=i,e^2=fh,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, 4990180);