Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 | C22×C4⋊C4 |
| prime radical | C23×D4 | Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1278 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)(9,10,15,19,13,16,14,24)(11,20,12,23,21,18,22,17);(1,2)(3,4)(6,8)(10,16)(11,21)(12,22)(19,24);(5,6,7,8)(9,11,15,12,13,21,14,22)(10,17,24,18,16,23,19,20);(1,3,4,2)(9,12,15,21,13,22,14,11)(10,18,24,23,16,20,19,17)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215055, #72215063, #72216201, … |
| maximal normal subgroup of | #72215055, #72215063, #72216201, … |
| maximal quotient of | #72214941, #72214947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294011);