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 | #37720 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,6,8,7)(17,18,23,24)(19,26,20,25)(21,29,31,27)(22,28,32,30);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,16,12)(18,24)(21,31)(25,26)(28,30);(4,8)(6,7)(10,14)(12,15)(17,19,23,20)(18,25,24,26)(21,28,31,30)(22,27,32,29);(2,3)(6,7)(10,14)(11,16)(17,20,23,19)(18,26,24,25)(21,28,31,30)(22,27,32,29);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,21,23,31)(18,27,24,29)(19,28,20,30)(22,25,32,26);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,22)(18,28)(19,29)(20,27)(21,25)(23,32)(24,30)(26,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8295157);