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 | |
6469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(10,13)(14,16)(17,18)(19,24)(20,23)(21,22);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,22,21)(19,23,20,24);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,22)(21,24);(3,7)(6,8)(17,18,22,21)(19,23,20,24);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,23,21,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,18)(19,23,20,24)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72261997, #72262018, #72280742, … |
| maximal normal subgroup of | #72261997, #72262018, #72280742, … |
| maximal quotient of | #72281263, #72281949, #72300253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842150);