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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6594 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)(13,14,18,19)(15,20,16,17);(1,2,4,3)(10,12)(14,19)(15,16);(5,6,8,7)(9,10)(11,12)(13,15,18,16)(14,17,19,20);(2,3)(5,7,8,6)(13,16,18,15)(14,17,19,20);(1,3,4,2)(13,17)(14,16)(15,19)(18,20)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=i,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 | #72232433, #72232450, #72263777, … |
| maximal normal subgroup of | #72232433, #72232450, #72263777, … |
| maximal quotient of | #72330274, #72330559, #72330889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 427512);