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 | |
5017 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)(3,4)(5,8)(6,7)(9,10,13,11)(12,15,16,14)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,3)(2,8)(4,6)(5,7)(10,11)(14,15)(18,22)(23,25)(24,29)(30,31);(1,2,7,6)(3,8,5,4)(9,11,13,10)(12,15,16,14)(19,26)(20,28)(23,25)(24,29);(1,4)(2,5)(3,6)(7,8)(9,12,13,16)(10,14,11,15)(17,19,21,26)(18,23,22,25)(20,27,28,32)(24,30,29,31);(1,5,7,3)(2,8,6,4)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32);(1,6,7,2)(3,8,5,4)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8680021);