Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1718 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)(7,8)(10,11)(13,14,17,15)(16,18,20,19)(21,22)(23,27)(24,26)(25,28)(29,34)(30,31)(32,33)(35,36);(5,6,9,12)(7,10,8,11)(13,15,17,14)(16,18,20,19)(22,24)(27,31)(28,32)(34,35);(5,7,9,8)(6,10,12,11)(13,16,17,20)(14,18,15,19)(21,23,26,30)(22,27,24,31)(25,29,33,36)(28,34,32,35);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(21,24,26,22)(23,27,30,31)(25,28,33,32)(29,35,36,34);(5,8,9,7)(6,11,12,10)(21,25)(22,28)(23,29)(24,32)(26,33)(27,34)(30,36)(31,35)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2644439);