Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2382 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)(8,9)(10,12)(14,17)(18,19)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,36,35,31);(6,9)(7,12)(13,14,16,17)(15,19,20,18)(22,27)(23,28)(25,34)(31,36);(5,6,8,9)(7,10,12,11)(13,14,16,17)(15,18,20,19)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,36)(34,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,33)(34,36);(5,7,8,12)(6,10,9,11)(13,15,16,20)(14,18,17,19)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,31,32,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4373611);