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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2060 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)(5,6,10,12)(7,8,9,11)(13,14,16,19)(15,17,20,18)(23,29)(24,31)(27,32)(28,33);(5,7)(6,11)(8,12)(9,10)(13,15)(14,17)(16,20)(18,19)(22,26)(27,32)(28,33)(34,36);(6,12)(7,9)(13,15,16,20)(14,18,19,17)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(1,2,3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(5,9,10,7)(6,8,12,11)(13,14)(15,18)(16,19)(17,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=g,c^2=fg,d^2=hi,e^2=fgh,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, 4991498);