Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129633 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,14,18,13,17,11,15)(12,20,19,23,22,24,21,16);(1,2,4,6)(3,7,8,5)(9,11,13,14)(10,15)(12,21,22,19)(16,20)(17,18)(23,24);(1,3,4,8)(2,5,6,7)(9,12,13,22)(10,16)(11,19,14,21)(15,20)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dh,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233008, #72233009, #72233141, … |
| maximal normal subgroup of | #72233008, #72233009, #72233141, … |
| maximal quotient of | #72233007, #72234061, #72236952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56685);