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 | |
4953 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)(3,5,4,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,28)(29,31)(30,32);(2,7)(3,4)(10,13)(14,15)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,31)(29,32);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,13)(15,16);(1,4,6,3)(2,5,7,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,29)(18,23,26,31)(19,27,25,32)(21,30,24,28);(1,5)(2,3)(4,7)(6,8)(17,19)(18,24)(20,27)(21,26)(22,25)(23,28)(29,32)(30,31);(17,21,22,24)(18,25,26,19)(20,28,29,30)(23,27,31,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595552);