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 | |
4671 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)(3,5)(4,6)(7,8)(9,10,13,11)(12,15,16,14)(18,23)(20,24)(21,29)(27,30);(2,3)(6,7)(10,11)(14,15)(18,23)(20,24)(26,31)(28,32);(9,11,13,10)(12,15,16,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,30)(29,31);(2,3)(4,8)(9,12,13,16)(10,14,11,15)(17,19,22,25)(18,24,23,20)(21,30,29,27)(26,28,31,32);(1,3,5,2)(4,7,8,6)(17,20,22,24)(18,25,23,19)(21,28,29,32)(26,30,31,27);(1,4)(2,6)(3,7)(5,8)(17,21,22,29)(18,26,23,31)(19,27,25,30)(20,28,24,32)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352169);