Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| 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 | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,13,12)(11,15,16,14)(18,21)(19,23)(25,26)(27,31)(28,29)(30,32);(2,6)(4,7)(9,11,13,16)(10,14,12,15)(18,21)(22,24)(26,29)(30,31);(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,4,6,7)(9,12,13,10)(11,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,27,28,32)(26,30,29,31);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232628, #72232679, #72232696, … |
| maximal normal subgroup of | #72232628, #72232679, #72232696, … |
| maximal quotient of | #72329928 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2730644);