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 | |
5277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,7,4,5)(9,10)(11,13)(12,15)(14,16)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(20,27)(24,30)(26,31)(28,32);(2,8)(3,4)(9,12,14,13)(10,15,16,11)(18,23)(19,25)(24,30)(26,31);(1,3,6,4)(2,5,8,7)(9,10)(11,13)(12,15)(14,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32);(1,4,6,3)(2,5,8,7)(9,10)(11,12)(13,15)(14,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32);(1,5)(2,4)(3,8)(6,7)(9,13,14,12)(10,11,16,15)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72265955, #72265960, #72265962 |
| maximal normal subgroup of | #72265955, #72265960, #72265962 |
| maximal quotient of | #72266510, #72266516 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9963014);