Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,8,10)(7,9,11,12)(14,17)(18,20)(21,22,26,23)(24,27,32,29)(25,28,33,30)(31,34,36,35);(5,7,8,11)(6,9,10,12)(15,19)(18,20)(22,23)(27,29)(28,30)(34,35);(5,7)(6,9)(8,11)(10,12)(13,14,16,17)(15,18,19,20)(21,23,26,22)(24,29,32,27)(25,28,33,30)(31,34,36,35);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,6)(7,9)(8,10)(11,12)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4514406);