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 | #39052 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,26)(19,21,24,20)(22,30,31,25)(27,28,32,29);(1,3,6,5)(2,7,4,8)(9,10,12,14)(11,13,16,15)(17,19,23,24)(18,20,26,21)(22,27,31,32)(25,28,30,29);(9,10)(11,13)(12,14)(15,16)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27);(1,2,6,4)(3,7,5,8)(9,11,12,16)(10,13,14,15)(17,21,23,20)(18,19,26,24)(22,29,31,28)(25,27,30,32);(1,4,6,2)(3,7,5,8)(17,22,23,31)(18,25,26,30)(19,27,24,32)(20,28,21,29);(1,5,6,3)(2,7,4,8)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413193);