Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,19)(20,22,24,23)(25,26,28,30)(27,29,31,32);(1,2,5,3)(4,7,8,6)(9,11)(10,15)(12,13)(14,16)(18,19)(20,24)(27,31)(29,32);(2,3)(6,7)(10,12)(11,16)(17,19,21,18)(20,22,24,23)(25,27)(26,29)(28,31)(30,32);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,20,21,24)(18,22,19,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,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=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393705);