Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(9,10,12,11)(13,14)(15,16)(17,18)(19,20);(6,8)(9,11,12,10)(14,18)(16,19);(1,2,4,3)(5,6,7,8)(14,18)(15,20);(1,3,4,2)(13,15,17,20)(14,19,18,16);(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=g,e^2=f,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 | C4×#45355, #72257918, #72257921, … |
| maximal normal subgroup of | C4×#45355, #72257918, #72257921, … |
| maximal quotient of | #72264429, #72264534 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 443264);