Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,23,24,20)(25,26,28,30)(27,31,32,29);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,12,14,15)(17,19)(18,20)(21,24)(22,23)(25,26)(27,31)(28,30)(29,32);(3,7)(6,8)(10,14)(12,15)(19,24)(20,23)(25,26,28,30)(27,29,32,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,26)(27,31)(28,30)(29,32);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27,28,32)(26,29,30,31)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=fhi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72263168, #72263193, #72263195, … |
| maximal normal subgroup of | #72263168, #72263193, #72263195, … |
| maximal quotient of | #72318658 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3165183);