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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2040 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)(7,9)(8,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35);(6,7)(8,12)(13,15)(14,18)(16,19)(17,20)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(5,6,9,7)(8,11,12,10)(13,14,17,19)(15,18,20,16)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16)(14,15)(17,18)(19,20)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,8,9,12)(6,10,7,11)(13,14)(15,16)(17,19)(18,20)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=f,d^2=i,e^2=h,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, 4989968);