Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(10,18)(13,20);(5,6)(7,13,14,20)(8,12)(9,11)(10,15,18,16)(17,19);(5,7)(6,10)(8,16)(9,14)(11,18)(12,20)(13,19)(15,17);(1,2,3,4)Polycyclic presentation:
2^9|b^2=e,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2010, #72237198, #72239252, … |
| maximal normal subgroup of | D4×#2010, #72237198, #72239252, … |
| maximal quotient of | #72226727, #72237249, #72237250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319062);