Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C4⋊C4 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,6,8,3)(9,10,13,18,14,16,12,17)(11,19,22,15,20,24,21,23);(2,6)(3,7)(9,11,12,22,14,20,13,21)(10,15,17,24,16,23,18,19);(1,3,8,6,4,7,5,2)(11,20)(15,23)(19,24)(21,22);(10,16)(15,23)(17,18)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,g^2=h,b^a=be,d^a=dh,c^b=ci,e^b=eh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286355, #72286428, #72286469, … |
| maximal normal subgroup of | #72286355, #72286428, #72286469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90706);