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 | |
5614 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)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32);(2,3)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31);(4,8)(6,7)(10,13)(14,15)(18,23)(22,25)(24,28)(31,32);(2,3)(6,7)(11,16)(14,15)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,14,13,15)(17,19)(18,22)(20,29)(21,27)(23,25)(24,32)(26,30)(28,31);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=h,e^2=ghi,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, 8269258);