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 | #35346 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,25)(20,24)(21,27)(22,23)(26,32)(28,29)(30,31);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,15,16,12)(17,19)(18,21)(20,28)(22,27)(23,25)(24,32)(26,30)(29,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(18,23)(21,25)(24,31)(29,32);(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,21)(20,28)(22,27)(23,25)(24,32)(26,30)(29,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72284296, #72284522, #72284596, … |
| maximal normal subgroup of | #72284296, #72284522, #72284596, … |
| maximal quotient of | #72333445, #72333473, #72334056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840071);