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 | |
5133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10,13,14)(11,12,15,16)(17,18,22,23)(19,20,26,24)(21,29,30,25)(27,31,32,28);(3,7)(6,8)(10,14)(11,15)(18,23)(19,26)(25,29)(27,32);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,15)(14,16)(18,23)(20,24)(25,29)(28,31);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32);(1,4,5,2)(3,8,7,6)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32);(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026472);