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×#3287 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,15)(11,13)(16,23)(17,21)(18,22)(19,24);(9,10,14,17)(11,15,13,21)(12,19,20,23)(16,22,24,18);(2,4)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19);(5,7,8,6)(10,17)(15,21)(16,24)(19,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=h,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 | #72214822, D4×#2943, #72282290, … |
| maximal normal subgroup of | #72214822, D4×#2943, #72282290, … |
| maximal quotient of | #72280320, #72287242, #72287475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291011);