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 | |
5018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,11)(12,15,16,14)(17,18,22,24)(19,23,26,20)(21,25,30,31)(27,29,32,28);(1,3,4,7)(2,5,6,8)(10,11)(14,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(18,24)(19,26)(25,31)(27,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,19)(21,29)(22,23)(24,26)(25,32)(27,31)(28,30);(1,2)(3,5)(4,6)(7,8)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32);(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424604);