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) = C29 |
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 | C2×#3038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,4,3)(5,6,8,7)(10,14)(11,12)(15,16);(1,3,4,2)(6,7);(9,10)(11,14)(12,16)(13,15);(5,7,8,6);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,g^c=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D4×AGL(1,ℤ/8), #72279507, #72279595, … |
| maximal normal subgroup of | C4×D4×AGL(1,ℤ/8), #72279507, #72279595, … |
| maximal quotient of | #72279507, #72279990, #72280295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289051);