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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2020 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)(21,22,26,30)(23,31,32,27)(24,33,35,28)(25,34,36,29);(8,12)(10,11)(13,15,17,20)(14,18,16,19)(23,32)(25,36)(27,31)(29,34);(6,7)(10,11)(13,14,17,16)(15,18,20,19)(21,23)(22,27)(24,34)(25,33)(26,32)(28,36)(29,35)(30,31);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,17,14)(15,18,20,19)(21,24,26,35)(22,28,30,33)(23,29,32,34)(25,31,36,27);(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,18)(17,20)(21,25)(22,29)(23,33)(24,31)(26,36)(27,35)(28,32)(30,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=i,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, 4990213);