Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,9,7)(8,11,12,10)(15,19)(18,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,34)(35,36);(5,6,9,7)(8,10,12,11)(14,17)(18,20)(22,27)(23,29)(25,33)(31,36);(2,3)(6,7)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,29)(22,24,27,32)(25,30,33,35)(28,31,34,36);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,19)(17,20)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,33)(22,28,27,34)(23,30,29,35)(24,31,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4856656);