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 | |
7061 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(5,8)(11,16)(12,14)(17,18)(19,24)(20,23)(21,22);(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(18,21)(20,24);(10,15)(11,16)(17,18,22,21)(19,24,23,20);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,20,22,24)(18,23,21,19);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,18)(19,24,23,20)Polycyclic presentation:
2^9|b^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72233544, #72233547, #72233598, … |
| maximal normal subgroup of | #72233544, #72233547, #72233598, … |
| maximal quotient of | #72263111, #72281715, #72297821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968296);