Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2040 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)(6,9)(7,11)(13,14,16,18)(15,17,19,20)(21,22,26,29)(23,30,27,24)(25,33,36,28)(31,32,34,35);(7,11)(10,12)(15,19)(17,20)(21,23,26,27)(22,24,29,30)(25,34,36,31)(28,35,33,32);(5,6,8,9)(7,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23)(22,24)(25,34)(26,27)(28,35)(29,30)(31,36)(32,33);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,27)(23,29)(25,35)(26,30)(28,31)(32,36)(33,34);(5,7,8,11)(6,10,9,12)(21,25)(22,28)(23,31)(24,32)(26,36)(27,34)(29,33)(30,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4039631);