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 | |
5428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(6,8)(10,12)(13,15)(17,18,22,23)(19,25,28,24)(20,21,29,30)(26,31,32,27);(2,5)(6,8)(10,12)(11,16)(18,23)(20,29)(24,25)(26,32);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,15,16,13);(9,11,14,16)(10,15,12,13)(17,19,22,28)(18,24,23,25)(20,26,29,32)(21,27,30,31);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,13,16,15)(17,20)(18,21)(19,26)(22,29)(23,30)(24,27)(25,31)(28,32);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,21)(18,20)(19,27)(22,30)(23,29)(24,26)(25,32)(28,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,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^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015526);