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 | |
4901 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,28,29,24)(20,30,32,25)(21,22,26,31);(1,3,6,4)(2,7,5,8)(9,10,12,14)(11,13,16,15)(17,19,23,29)(18,24,27,28)(20,31,32,22)(21,30,26,25);(1,3)(2,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(20,32)(21,26)(22,31)(25,30);(3,4)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,22)(21,28)(23,32)(24,26)(27,30)(29,31);(1,4,6,3)(2,8,5,7)(17,21,23,26)(18,22,27,31)(19,25,29,30)(20,24,32,28);(1,5,6,2)(3,8,4,7)(17,22)(18,26)(19,20)(21,27)(23,31)(24,25)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970508);