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 | |
5220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(3,7)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32);(2,5)(3,7)(10,11)(14,15)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(19,27)(20,29)(23,28)(24,32);(2,5)(6,8)(9,12,13,16)(10,14,11,15)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25);(1,2)(3,6)(4,5)(7,8)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29);(1,3)(2,6)(4,7)(5,8)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8744530);