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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2538 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)(8,12)(9,11)(13,14,18,20)(15,16,19,17);(5,7,9,8)(6,10,11,12)(14,15)(16,17)(19,20);(5,8,9,7)(6,10,11,12);(13,15)(14,16)(17,20)(18,19);(13,16,18,17)(14,19,20,15)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72305395, #72305404, #117×AGL(1,ℤ/8) |
| maximal normal subgroup of | #72305395, #72305404, #117×AGL(1,ℤ/8) |
| maximal quotient of | #72305719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287708);