Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C2×#2408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4538 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)(11,12)(13,14)(15,16)(17,18)(19,21)(20,23)(22,24);(1,2,4,3)(5,6,7,8)(10,14)(11,16)(17,19,22,23)(18,20,24,21);(5,6)(7,8)(10,14)(12,15)(17,20,22,21)(18,23,24,19);(1,3,4,2)(9,11,13,16)(10,15,14,12)(17,21,22,20)(18,23,24,19);(9,12,13,15)(10,16,14,11)(17,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fg,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72261162, #72261565, #72262153, … |
| maximal normal subgroup of | #72261162, #72261565, #72262153, … |
| maximal quotient of | #72277984, C2×#579825, #72299587, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 487021);