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 | #45999 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4529 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,30)(31,32);(3,8)(6,7)(9,10,12,14)(11,13,16,15)(17,19,23,21)(18,24,27,28)(20,29,26,31)(22,30,25,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,26)(18,25,27,22)(19,29,21,31)(24,32,28,30);(1,2,5,4)(3,7,8,6)(9,11)(10,13)(12,16)(14,15);(1,3)(2,7)(4,6)(5,8)(17,21,23,19)(18,24,27,28)(20,31,26,29)(22,30,25,32);(1,4,5,2)(3,7,8,6)(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,31,28,29)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219281, #72219283 |
| maximal quotient of | #72219271 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620943);