Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,11)(12,16)(17,18)(19,24)(20,21)(22,23)(26,29)(30,32);(2,5)(6,7)(9,10,13,11)(12,14,16,15)(17,19)(18,21)(20,22)(23,24)(25,26)(27,30)(28,29)(31,32);(2,5)(6,7)(10,11)(14,15)(18,23)(19,20)(27,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,19)(18,21,23,24)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,21)(18,19)(20,23)(22,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|b^2=i,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72258910, #72258923, #72266556, … |
| maximal normal subgroup of | #72258910, #72258923, #72266556, … |
| maximal quotient of | #72337170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 831813);