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 | |
5913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,16)(12,15)(13,14)(18,22)(23,26)(24,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10,14,13)(11,15,12,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(2,5)(3,7)(9,11)(10,15)(12,14)(13,16)(19,25)(20,28)(23,26)(24,29);(3,7)(6,8)(9,10,14,13)(11,15,12,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3)(2,6)(4,7)(5,8)(9,12,14,11)(10,15,13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,15,12,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72261791, #72261985, #72262049, … |
| maximal normal subgroup of | #72261791, #72261985, #72262049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842538);