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 | #35703 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(18,20)(19,25)(21,29)(30,32);(2,6)(4,7)(17,18,22,20)(19,26,25,23)(21,28,29,24)(27,30,31,32);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(1,3)(2,4)(5,8)(6,7)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,4)(2,3)(5,7)(6,8)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7712062);