Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397902 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,14)(8,19)(9,12)(10,13)(11,16)(15,17)(18,20);(5,7,8,16,10,17,9,18)(6,11,12,14,13,20,19,15);(5,6)(7,15,17,14)(8,19)(9,12)(10,13)(11,16,20,18);(2,4)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2010, #72237198, #72239534, … |
| maximal normal subgroup of | D4×#2010, #72237198, #72239534, … |
| maximal quotient of | #72237215, #72237217, #72237253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305144);