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 |
| prime radical | #37645 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,3)(6,7)(9,10,13,15)(11,12,16,14)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31);(1,2,5,3)(4,6,8,7)(9,11,13,16)(10,14,15,12)(19,28)(20,29)(24,30)(25,32);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32);(1,2,5,3)(4,7,8,6)(9,12)(10,11)(13,14)(15,16)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31);(1,3,5,2)(4,6,8,7)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32);(1,4)(2,6)(3,7)(5,8)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72317544, #72317587, #72317637, … |
| maximal normal subgroup of | #72317544, #72317587, #72317637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8603447);