Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,13,10,14,8,11)(7,15,18,20,16,19,17,12);(1,2)(3,4)(6,11,14,13)(7,16)(12,15,20,19)(17,18);(2,4)(5,6)(7,12)(8,13)(9,11)(10,14)(15,17)(16,20)(18,19);(5,7,9,18,10,16,8,17)(6,12,11,19,14,20,13,15)Polycyclic presentation:
2^8|a^2=f,b^2=ef,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #502213, #504844, #504886, … |
| maximal normal subgroup of | #502213, #504844, #504886, … |
| maximal quotient of | #288644, #288647, #288650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18432);