Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| 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 | C42×D4 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23);(2,3)(7,8)(9,11,13,10,14,16,18,17)(12,19,20,15,21,22,24,23);(2,3)(7,8)(9,12,13,20,14,21,18,24)(10,15,16,22,17,23,11,19);(1,3,5,8,4,2,6,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ci,e^a=eh,d^b=di,e^b=eh,d^c=di140 conjugacy classes — view the full character table.
| maximal subgroup of | #72286221, #72286245, #72286355, … |
| maximal normal subgroup of | #72286221, #72286245, #72286355, … |
| maximal quotient of | #72319249, #72320674 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91241);