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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,8)(3,7,4,5)(9,10,13,12)(11,14,16,15)(17,18)(19,24)(20,22)(21,23);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,19,21,20)(18,22,23,24);(2,8)(3,4)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,24)(21,23);(1,3,6,4)(2,5,8,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,24)(20,22)(21,23);(1,4,6,3)(2,5,8,7)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,22,23,24);(1,5)(2,4)(3,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72234871, #72234881, #72278365, … |
| maximal normal subgroup of | #72234871, #72234881, #72278365, … |
| maximal quotient of | #72261621, #72278272, #72278309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9976128);