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 | |
2189 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)(5,6,9,10)(7,8,11,12)(13,14,17,19)(15,18,20,16)(21,22)(23,29)(24,26)(25,32)(27,30)(28,33)(31,35)(34,36);(6,10)(7,11)(13,15,17,20)(14,16,19,18)(22,24)(23,30)(25,33)(35,36);(5,7,9,11)(6,8,10,12)(13,15,17,20)(14,18,19,16)(21,23)(22,27)(24,29)(25,34)(26,30)(28,36)(31,33)(32,35);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,36,34,35);(5,8)(6,7)(9,12)(10,11)(13,16,17,18)(14,15,19,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,35)(29,36)(30,34)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fg,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, 4990442);