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 | |
4767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,12,16,14)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,15,12);(2,4)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30);(1,4,5,2)(3,7,8,6)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32);(1,3)(2,6)(4,7)(5,8)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10243973);