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 | |
5104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,7)(3,8,4,5)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(1,3,6,4)(2,5,7,8)(9,10,13,12)(11,14,15,16)(20,28)(23,30)(26,32)(29,31);(2,7)(5,8)(10,12)(14,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(2,7)(3,4)(11,15)(14,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(1,4,6,3)(2,8,7,5)(9,11,13,15)(10,14,12,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(1,5,6,8)(2,3,7,4)(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8249019);