Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511853 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32899 |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,9)(4,6)(5,8)(7,11)(10,15)(12,14)(13,16);(2,6)(7,13)(8,14)(9,15);(4,12)(10,16);(1,3,5,11)(2,7)(4,10,12,16)(6,13)(8,9)(14,15)Polycyclic presentation:
2^9|d^2=h,b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,e^c=eh,g^d=gi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72216366, #72233206, C2×#511853, … |
| maximal normal subgroup of | #72216366, #72233206, C2×#511853, … |
| maximal quotient of | #72232813, #72232947, #72233009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419049);