Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#459 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,22)(21,24);(1,2,4,3)(10,14)(11,18)(13,20)(15,23)(17,24)(19,21);(6,8)(9,11,12,19)(10,15,14,17)(13,21,20,18)(16,23,22,24);(1,3,4,2)(10,14)(11,19)(16,22)(18,21)Polycyclic presentation:
2^8|a^2=fh,b^2=eh,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #273507, #273524, #273543, … |
| maximal normal subgroup of | #273507, #273524, #273543, … |
| maximal quotient of | #277109, #277110, #277111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17737);