Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,20,22,16)(13,17)(14,18)(19,24,21,23);(1,2,5,7)(3,6,4,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(1,3)(4,5)(6,8)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72268288, #72268295, #72268320, … |
| maximal normal subgroup of | #72268288, #72268295, #72268320, … |
| maximal quotient of | #72313296, #99×#106 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127491);