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 | |
5356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,25,26,24)(20,29,31,21)(27,32,30,28);(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,16,15)(18,23)(21,29)(24,25)(28,32);(1,3,5,4)(2,6,8,7)(9,10)(11,13)(12,14)(15,16)(19,26)(24,25)(27,30)(28,32);(3,4)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,26)(18,24,23,25)(20,30,31,27)(21,28,29,32);(1,4,5,3)(2,7,8,6)(17,20)(18,21)(19,27)(22,31)(23,29)(24,32)(25,28)(26,30);(1,2)(3,6)(4,7)(5,8)(17,21)(18,20)(19,28)(22,29)(23,31)(24,30)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72326379 |
| maximal normal subgroup of | #72326379 |
| maximal quotient of | #72326400, #72326740, #72327630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8751702);