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 | |
5014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(4,8)(6,7)(10,12)(14,16)(18,23)(19,27)(21,25)(28,32);(1,2,5,3)(4,6,8,7)(11,15)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32);(2,3)(6,7)(9,10,13,12)(11,14,15,16)(19,27)(24,30)(28,32)(29,31);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,31)(26,32);(9,12,13,10)(11,16,15,14)(17,20,22,26)(18,25,23,21)(19,28,27,32)(24,31,30,29);(1,4)(2,6)(3,7)(5,8)(17,21,22,25)(18,26,23,20)(19,29,27,31)(24,32,30,28)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014737);