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 | |
4756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,22)(21,28)(23,26)(24,29)(27,30)(31,32);(1,3,5,4)(2,6,7,8)(9,10,12,14)(11,13,16,15)(18,20)(19,26)(24,28)(27,32);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,2)(3,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30);(1,2)(3,6)(4,8)(5,7)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31);(1,4,5,3)(2,6,7,8)Polycyclic presentation:
2^9|b^2=gh,d^2=i,e^2=h,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, 8595590);