Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #32926 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2539 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 2539 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(3,7)(4,8)(13,16);(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,15)(14,16);(1,3,5,2)(4,6,8,7)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh65 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×B(4,𝔽2), #72241527, #72242976, … |
| maximal normal subgroup of | SD16×B(4,𝔽2), #72281249, #72313072, … |
| maximal quotient of | #72252889, #72281274, #72281461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253717);