Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(6,8)(10,14)(12,16)(18,22)(19,25)(20,28)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30);(2,5)(6,8)(9,11,13,15)(10,12,14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72267544 |
| maximal normal subgroup of | #72267544 |
| maximal quotient of | #72267545, #72267548, #72267626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8032903);