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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16250 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2521 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)(7,12)(8,11)(9,10)(13,14)(15,18)(16,20)(17,19)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(6,10)(7,11)(13,15,17,20)(14,16,19,18)(24,32)(27,34)(30,36)(33,35);(7,11)(8,12)(15,20)(16,18)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2,3,4)(5,7,9,11)(6,8,10,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,8)(6,11)(7,10)(9,12)(13,16,17,18)(14,15,19,20)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=h,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, 4990291);