Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,17)(15,20)(18,19);(13,14,16,18)(15,19,20,17);(6,9)(10,12)(14,18)(17,19);(14,19)(15,20)(17,18);(5,7,8,11)(6,10,9,12)(14,18)(17,19)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72240753, #72250423, #72256875, … |
| maximal normal subgroup of | #72240753, #72250423, #72256875, … |
| maximal quotient of | #72240775, #72305529, #72308231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288122);