Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12)(7,11,8,10)(13,14,17,20)(15,18,23,27)(16,19,24,28)(21,25,22,26);(5,7,9,8)(6,10,12,11)(13,15,16,22,17,23,24,21)(14,18,19,26,20,27,28,25);(1,2,3,4)(5,7,9,8)(6,11,12,10)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(5,8,9,7)(6,11,12,10)Polycyclic presentation:
2^8|a^2=eh,b^2=eg,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #122751, #122767, #122827, … |
| maximal normal subgroup of | #122751, #122767, #122827, … |
| maximal quotient of | #325562, #325563, #325564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21725);