Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,8)(13,14)(15,18)(16,19)(17,20);(1,2)(3,4)(9,10)(11,12);(6,8)(10,12)(14,18)(15,16)(19,20);(2,4)(14,19)(15,16)(18,20)Polycyclic presentation:
2^9|f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239715, #72239726, #72239750, … |
| maximal normal subgroup of | #72239715, #72239726, #72239750, … |
| maximal quotient of | #72250621, #72257250, #72320314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324777);