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 | |
4882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,14,15)(11,13,12,16)(17,18,22,24)(19,25,27,20)(21,23,29,30)(26,31,32,28);(1,2)(3,4)(5,6)(7,8)(10,15)(11,12)(17,18)(19,25)(20,27)(21,23)(22,24)(26,31)(28,32)(29,30);(2,6)(4,7)(9,11,14,12)(10,16,15,13)(21,29)(23,30)(26,32)(28,31);(2,6)(3,8)(9,12,14,11)(10,13,15,16)(17,19,22,27)(18,20,24,25)(21,26,29,32)(23,28,30,31);(1,3)(2,7)(4,6)(5,8)(9,13)(10,11)(12,15)(14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(1,4,5,7)(2,3,6,8)(17,21,22,29)(18,23,24,30)(19,26,27,32)(20,28,25,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8305771);