Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10,14,15)(11,16,12,13)(17,18,22,20)(19,24,21,23);(6,8)(10,15)(11,12)(17,19,22,21)(18,23,20,24);(5,6)(7,8)(9,11,14,12)(10,13,15,16)(17,18,22,20)(19,23,21,24);(1,3,4,2)(9,12,14,11)(10,13,15,16)(17,20,22,18)(19,23,21,24);(9,13)(10,12)(11,15)(14,16)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef140 conjugacy classes — view the full character table.
| maximal subgroup of | #72279950, #72280046, #72280386, … |
| maximal normal subgroup of | #72279950, #72280046, #72280386, … |
| maximal quotient of | C2×#519348, #72330527, #72331327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426544);