Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2408 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,8)(9,10)(11,12)(13,14,15,19,16,20,17,18);(1,2)(3,4)(9,11,12,10)(13,15,16,17)(14,18,20,19);(2,4)(10,11)(14,19)(15,17)(18,20);(6,8)(10,11)Polycyclic presentation:
2^9|a^2=ef,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dei,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239716, #72239726, #72239733, … |
| maximal normal subgroup of | #72239716, #72239726, #72239733, … |
| maximal quotient of | #72250324, #72257262, #72320481 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326332);