Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
607 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 607 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,3,4)(9,10,13,16,14,17,12,21)(11,15,19,22,20,23,18,24);(5,6)(7,8)(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,24)(20,23);(2,4)(6,8)(9,11,14,20)(10,15,17,23)(12,18,13,19)(16,22,21,24);(2,4)(9,12)(11,18)(13,14)(16,21)(19,20)(22,24)Polycyclic presentation:
2^8|a^2=ef,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #123093, #123103, #6634524, … |
| maximal normal subgroup of | #123093, #123103, #6634524, … |
| maximal quotient of | #343066, #343067, #343068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24054);