Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,8,7)(13,14,16,17)(15,19,20,18);(9,10,11,12)(13,14)(15,19)(16,17)(18,20);(1,2,4,3)(5,7,8,6)(9,10)(11,12)(14,17)(18,19);(1,3,4,2)(13,14)(15,18)(16,17)(19,20);(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=gi,d^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72232084, #72232094, #72232121, … |
| maximal normal subgroup of | #72232084, #72232094, #72232121, … |
| maximal quotient of | C2×#536623, #72330911, #72331073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 443819);