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 | |
4835 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)(9,10)(11,15)(12,13)(14,16)(17,18,22,24)(19,26,29,23)(20,21,31,25)(27,28,30,32);(1,3,5,7)(2,4,6,8)(10,13)(14,15)(17,19,22,29)(18,23,24,26)(20,30,31,27)(21,28,25,32);(9,10)(11,14)(12,13)(15,16)(18,24)(20,31)(23,26)(27,30);(2,6)(3,7)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,27)(21,24)(22,31)(23,28)(26,32)(29,30);(1,2)(3,4)(5,6)(7,8)(17,19)(18,26)(20,30)(21,32)(22,29)(23,24)(25,28)(27,31);(1,4)(2,7)(3,6)(5,8)(17,21,22,25)(18,20,24,31)(19,28,29,32)(23,30,26,27)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154588);