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 | #43921 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,5)(3,7,8,4)(9,10,13,12)(11,15,16,14);(1,3,6,8)(2,7,5,4)(9,11,13,16)(10,14,12,15)(17,18)(19,26)(20,23)(21,29)(22,25)(24,31)(27,28)(30,32);(2,5)(3,8)(17,19,22,23)(18,20,25,26)(21,30,31,27)(24,32,29,28);(1,2,6,5)(3,7,8,4)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32);(1,4,6,7)(2,8,5,3)(9,10,13,12)(11,14,16,15)(17,20)(18,23)(19,25)(21,28)(22,26)(24,30)(27,29)(31,32);(1,5,6,2)(3,7,8,4)(9,12,13,10)(11,15,16,14)(17,21,22,31)(18,24,25,29)(19,27,23,30)(20,28,26,32)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7750760);