Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1760 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 1760 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 12 points:
(9,12);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12);(2,4)(5,7)(6,9)(8,11)(10,12);(5,6)(7,9)(8,10)(11,12)Polycyclic presentation:
2^8|b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fh,e^c=eh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139700, #139702, #139731, … |
| maximal normal subgroup of | #139700, #139702, #139731, … |
| maximal quotient of | #245225, #245226, #245227, … |
| automorphism group of | #117, #124, #323, … |
| outer automorphism group of | #493, #494, #1543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16888);