Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2867 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 163 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,12)(7,11,8,10)(14,17)(15,18)(19,20);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,14,16,17)(15,18,19,20);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,20,16,19,14,18);(5,8,9,7)(6,11,12,10)(13,14,16,17)(15,18,19,20)Polycyclic presentation:
2^8|a^2=eh,b^2=eg,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #101723, #101743, #101843, … |
| maximal normal subgroup of | #101723, #101743, #101843, … |
| maximal quotient of | #325558, #325563, #325568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21737);