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 | #2539 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,5,7,4)(10,13)(14,16);(1,3,6,7)(2,4,8,5)(9,10,12,13)(11,14,15,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,28)(31,32);(1,2)(3,4)(5,7)(6,8)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32);(1,4,6,5)(2,3,8,7)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32);(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32);(1,5,6,4)(2,7,8,3)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7539198);