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 | |
5063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,15)(14,16);(2,3)(6,7)(10,11)(14,15)(17,18,22,23)(19,24,28,30)(20,27,31,25)(21,26,32,29);(4,8)(6,7)(9,10,13,11)(12,14,16,15)(18,23)(21,32)(24,30)(25,27);(2,3)(6,7)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31);(1,3,5,2)(4,7,8,6)(9,12,13,16)(10,14,11,15)(17,20,22,31)(18,25,23,27)(19,26,28,29)(21,30,32,24);(1,4,5,8)(2,6,3,7)(17,21)(18,26)(19,27)(20,30)(22,32)(23,29)(24,31)(25,28)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8273826);