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) = C24×C42 |
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 | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(10,11)(12,16)(18,22)(19,24)(25,26,28,29)(27,31,32,30);(2,6)(3,7)(9,10,13,11)(12,14,16,15)(17,18)(19,23)(20,24)(21,22)(26,29)(30,31);(2,6)(4,8)(12,16)(14,15)(18,22)(20,23)(25,27,28,32)(26,30,29,31);(1,3)(2,4)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,23,22,20)(25,26)(27,30)(28,29)(31,32);(1,4)(2,3)(5,8)(6,7)(9,12,13,16)(10,14,11,15)(17,20,21,23)(18,24,22,19)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fgi,d^2=fg,e^2=fg,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, 4540026);