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 |
| prime radical | #37836 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4886 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)(10,13)(11,15)(18,23)(20,27)(21,29)(26,31);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(18,23)(19,25)(20,27)(28,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32);(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32);(1,4)(2,6)(3,7)(5,8)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)Polycyclic presentation:
2^9|b^2=g,c^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017125);