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 | |
5669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(3,8)(9,10,13,15)(11,14,16,12)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,30)(18,23,26,27)(20,21,29,28)(24,25,31,32);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,32,27,25);(1,3,5,8)(2,7,6,4)(9,11,13,16)(10,14,15,12)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31);(1,4,5,7)(2,3,6,8)(9,12,13,14)(10,11,15,16)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72265954, #72265955, #72265956, … |
| maximal normal subgroup of | #72265954, #72265955, #72265956, … |
| maximal quotient of | #72266531, #72266570 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627979);