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 | |
4938 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)(3,8)(4,7)(5,6)(10,12)(13,15)(17,18)(19,27)(20,30)(21,23)(22,28)(24,29)(25,31)(26,32);(2,6)(7,8)(10,12)(11,16)(18,21)(20,31)(24,27)(26,28);(9,10,14,12)(11,15,16,13)(17,19)(18,24)(20,28)(21,27)(22,25)(23,29)(26,31)(30,32);(1,3,5,4)(2,7,6,8)(9,11,14,16)(10,15,12,13)(17,20)(18,25)(19,28)(21,30)(22,24)(23,31)(26,29)(27,32);(1,2)(3,7)(4,8)(5,6)(9,12,14,10)(11,13,16,15)(17,21,23,18)(19,27,29,24)(20,30,31,25)(22,28,32,26);(1,4,5,3)(2,8,6,7)(9,13)(10,11)(12,16)(14,15)(17,22,23,32)(18,26,21,28)(19,25,29,30)(20,24,31,27)Polycyclic presentation:
2^9|c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015946);