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 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,14)(13,16)(18,21)(22,24);(1,3,5,7)(2,6,8,4)(9,11)(10,13)(12,16)(14,15)(18,22)(20,23)(21,24);(1,4,5,6)(2,7,8,3)(10,14)(13,15)(17,18,19,21)(20,24,23,22);(2,8)(3,7)(11,16)(13,15)Polycyclic presentation:
2^8|a^2=efh,b^2=efh,c^2=ef,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142491, #142507, #143019, … |
| maximal normal subgroup of | #142491, #142507, #143019, … |
| maximal quotient of | #384613, #384615, #386810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25456);