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 | |
4971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,24)(19,25,20,21)(22,26,30,31)(27,32,28,29);(10,13)(14,15)(18,24)(21,25)(26,31)(29,32);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,19)(18,25,24,21)(22,27,30,28)(26,29,31,32);(1,3)(2,6)(4,8)(5,7)(17,21,23,25)(18,20,24,19)(22,29,30,32)(26,28,31,27);(1,4,5,2)(3,8,7,6)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=gi,c^2=ghi,d^2=gi,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, 10352165);