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 | |
2168 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)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22,27,26)(23,29,24,28)(25,33,34,30)(31,35,32,36);(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,23,27,24)(22,28,26,29)(25,32,34,31)(30,36,33,35);(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,20,19)(21,24,27,23)(22,29,26,28)(25,32,34,31)(30,36,33,35);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,30)(23,31)(24,32)(26,33)(27,34)(28,35)(29,36);(5,7,8,11)(6,9,10,12)(13,15,16,20)(14,17,18,19)(21,26,27,22)(23,28,24,29)(25,33,34,30)(31,35,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=fi,e^2=g,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, 4987798);