Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,22)(26,27)(28,32);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24)(26,27)(30,31);(3,7)(6,8)(10,12)(14,15)(18,21)(19,24)(25,26,29,27)(28,30,32,31);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,24)(22,23)(25,27,29,26)(28,31,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,23,24,20)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=hi,c^2=fghi,d^2=fghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240130, #72240186, #72240188, … |
| maximal normal subgroup of | #72240130, #72240186, #72240188, … |
| maximal quotient of | #72251703, #72328869, #72328887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3763061);