Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×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 | |
3083 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,4)(2,6,7,8)(9,10,12,14)(11,13,16,15)(17,19,21,24)(18,22,23,20)(26,28)(27,31);(3,4)(6,8)(11,16)(13,15)(18,23)(20,22)(25,26,29,28)(27,32,31,30);(1,4,5,3)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,21,22)(18,19,23,24)(25,27,29,31)(26,30,28,32);(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|b^2=ghi,c^2=fghi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72262655, #72262673, #72262674, … |
| maximal normal subgroup of | #72262655, #72262673, #72262674, … |
| maximal quotient of | #72330401 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1445687);