Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
2021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,7)(5,6)(11,15)(14,16)(17,18,21,22)(19,23,24,20)(26,29)(27,31);(1,3,6,4)(2,7,5,8)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(26,29)(30,32);(2,5)(7,8)(18,22)(20,23)(25,26)(27,30)(28,29)(31,32);(1,4,6,3)(2,8,5,7)(9,11,13,15)(10,14,12,16)(17,19,21,24)(18,23,22,20)(25,26)(27,30)(28,29)(31,32);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,16,15,14)(17,20,21,23)(18,24,22,19)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,d^2=hi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2493660);