Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,8)(9,10,15,11,14,18,13,17)(12,16,23,19,22,20,21,24);(1,2,4,3)(9,11,15,18,14,17,13,10)(12,20,21,19,22,16,23,24);(5,6,7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(1,3,4,2)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,24,20,19)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72309811, #72309818, D4×#1946, … |
| maximal normal subgroup of | #72309811, #72309818, D4×#1946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269411);