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 |
| prime radical | #44927 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4983 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)(5,7)(6,8)(17,18)(19,24)(20,26)(21,27)(22,30)(23,25)(28,32)(29,31);(1,3,5,8)(2,6,7,4)(9,10,13,14)(11,15,16,12)(17,19)(18,24)(20,23)(21,28)(22,29)(25,26)(27,32)(30,31);(1,2,5,7)(3,6,8,4)(9,10)(11,15)(12,16)(13,14)(18,25)(19,20)(21,31)(22,32);(2,7)(3,8)(10,14)(11,16)(17,20,23,19)(18,26,25,24)(21,28,31,30)(22,27,32,29);(1,2)(3,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,21,23,31)(18,27,25,29)(19,28,20,30)(22,24,32,26);(1,4,5,6)(2,8,7,3)(9,12)(10,11)(13,15)(14,16)(17,22)(18,28)(19,29)(20,27)(21,24)(23,32)(25,30)(26,31)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=gh,e^2=gh,f^2=g,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, 8968626);