Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3690 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,29)(30,31);(2,5)(6,8)(10,14)(12,15)(17,19)(18,21)(20,24)(22,23)(25,27,30,28)(26,29,31,32);(3,7)(6,8)(10,14)(11,16)(19,24)(21,22)(26,31)(27,28);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,27)(26,29,31,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,27)(28,31)(30,32)Polycyclic presentation:
2^9|b^2=gi,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72326549, #72326596, #72326628, … |
| maximal normal subgroup of | #72326549, #72326596, #72326628, … |
| maximal quotient of | #72328561 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1025389);