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 | |
4773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,15)(12,14)(13,16);(1,2)(3,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27);(9,10,13,16)(11,14,12,15)(18,23)(21,31)(24,32)(25,28);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,26)(21,28)(22,29)(23,32)(25,31)(27,30);(1,3,5,4)(2,6,8,7)(9,12,13,11)(10,15,16,14)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32);(1,4,5,3)(2,7,8,6)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7675278);