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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1772 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)(15,19)(18,20)(21,22,26,30)(23,28,24,27)(25,33,34,29)(31,35,32,36);(5,7)(6,11)(8,10)(9,12)(14,17)(18,20)(21,23,26,24)(22,27,30,28)(25,32,34,31)(29,36,33,35);(7,8)(9,11)(13,14,16,17)(15,18,19,20)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(1,2,3,4)(5,8,10,7)(6,11,12,9)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33);(5,9,10,11)(6,7,12,8)(13,15)(14,18)(16,19)(17,20)(21,22)(23,28)(24,27)(25,29)(26,30)(31,36)(32,35)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=fi,e^2=fg,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, 5187031);