Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1910 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)(14,18)(15,16)(21,22,26,29)(23,25,30,27)(24,32,35,28)(31,36,33,34);(8,12)(10,11)(13,14,17,18)(15,19,16,20)(23,30)(25,27)(31,33)(34,36);(6,7)(10,11)(13,15,17,16)(14,19,18,20)(21,23)(22,27)(24,33)(25,29)(26,30)(28,34)(31,35)(32,36);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,17,15)(14,19,18,20)(21,24,26,35)(22,28,29,32)(23,31,30,33)(25,34,27,36);(5,8)(6,10)(7,11)(9,12)(13,14)(15,20)(16,19)(17,18)(21,25,26,27)(22,23,29,30)(24,34,35,36)(28,31,32,33)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5189473);