Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1993 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 1993 conjugacy classes of subgroups — too many to draw. Only its 303 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,13)(8,16)(9,11)(10,14)(12,18)(15,20)(17,19);(5,7)(6,10)(8,17)(9,15)(11,19)(12,20)(13,16)(14,18);(8,18)(12,16)(13,20)(14,17);(6,11)(10,19)(12,16)(13,20)Polycyclic presentation:
2^8|a^2=g,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=eh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147752, #359536, #359558, … |
| maximal normal subgroup of | #147752, #359536, #359558, … |
| maximal quotient of | #184930, #184936, #184942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13604);