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 | |
6138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(11,15)(13,16)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,24)(20,28)(21,26)(22,29)(23,27)(25,32)(30,31);(2,7)(4,6)(9,11,12,15)(10,13,14,16)(19,27)(21,24)(28,32)(29,30);(1,3,5,8)(2,6,7,4)(9,10,12,14)(11,13,15,16)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72317873, #72319776, #72319886, … |
| maximal normal subgroup of | #72317873, #72319776, #72319886, … |
| maximal quotient of | #72326795, #72326809, #72326872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8495177);