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 | #17086 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2356 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)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,27)(23,28,29,24)(25,30,34,35)(31,36,33,32);(5,6)(7,11)(8,10)(9,12)(13,15,16,19)(14,17,18,20)(22,27)(24,28)(30,35)(32,36);(5,6,8,10)(7,9,12,11)(13,14,16,18)(15,17,19,20)(21,23)(22,28)(24,27)(25,33)(26,29)(30,32)(31,34)(35,36);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,28)(22,29,27,23)(25,32,34,36)(30,33,35,31);(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,19)(18,20)(21,25)(22,30)(23,31)(24,32)(26,34)(27,35)(28,36)(29,33)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=g,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, 4990452);