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 | |
5432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,6,8,4)(10,13)(14,16)(18,23)(20,29)(24,27)(26,31);(1,3,5,8)(2,6,7,4)(11,15)(14,16)(17,18,22,23)(19,27,28,24)(20,30,29,25)(21,26,32,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(19,28)(21,32)(24,27)(26,31);(2,7)(3,8)(17,19)(18,24)(20,31)(21,25)(22,28)(23,27)(26,29)(30,32);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,22,29)(18,25,23,30)(19,26,28,31)(21,24,32,27);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,32)(18,26,23,31)(19,25,28,30)(20,24,29,27)Polycyclic presentation:
2^9|a^2=i,b^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841588);