Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
2220 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)(16,17)(18,20)(21,22,27,30)(23,28,24,26)(25,33,35,29)(31,34,32,36);(8,12)(10,11)(13,14)(15,19)(16,17)(18,20)(21,23,27,24)(22,26,30,28)(25,32,35,31)(29,36,33,34);(6,7)(10,11)(14,17)(18,19)(21,24,27,23)(22,28,30,26)(25,32,35,31)(29,36,33,34);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,20)(17,19)(21,26,27,28)(22,24,30,23)(25,34,35,36)(29,32,33,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5184378);