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 | #43527 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(4,8)(6,7)(9,10,12,14)(11,13,15,16)(17,18,23,25)(19,27,30,24)(20,21,31,32)(22,26,29,28);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,30)(18,24,25,27)(20,28,31,26)(21,29,32,22);(2,3)(6,7)(18,25)(20,31)(24,27)(26,28);(17,20)(18,21)(19,28)(22,27)(23,31)(24,29)(25,32)(26,30);(1,3,5,2)(4,7,8,6)(17,21)(18,20)(19,29)(22,30)(23,32)(24,28)(25,31)(26,27);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,22,23,29)(18,26,25,28)(19,21,30,32)(20,27,31,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7625294);