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 | |
1752 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)(7,8)(9,11)(10,12)(13,14,18,17)(15,20,16,19)(21,22)(23,28)(24,32)(25,30)(26,27)(29,33)(31,35)(34,36);(5,7,9,12)(6,10,11,8)(13,14)(15,19)(16,20)(17,18)(22,27)(23,30)(24,33)(35,36);(5,7,9,12)(6,8,11,10)(13,15,18,16)(14,19,17,20)(21,23,26,30)(22,28,27,25)(24,34,33,31)(29,35,32,36);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,16,18,15)(14,19,17,20)(21,24)(22,29)(23,31)(25,35)(26,33)(27,32)(28,36)(30,34);(5,8)(6,7)(9,10)(11,12)(13,17,18,14)(15,19,16,20)(21,25,26,28)(22,23,27,30)(24,35,33,36)(29,31,32,34)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=ghi,e^2=gh,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, 4992142);