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 | |
4939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,27,28,24)(20,21,26,25)(29,30,31,32);(1,2,5,4)(3,7,8,6)(11,15)(13,16)(18,23)(19,28)(20,26)(30,32);(2,4)(6,7)(9,10)(11,13)(12,14)(15,16)(18,23)(21,25)(24,27)(30,32);(3,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,31)(21,30)(22,28)(23,27)(25,32)(26,29);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,26)(18,25,23,21)(19,29,28,31)(24,30,27,32);(1,4,5,2)(3,7,8,6)(17,21,22,25)(18,26,23,20)(19,30,28,32)(24,29,27,31)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=g,f^2=gi,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, 8281458);