Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
505 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11,12,20,14,19,13,21)(10,15,16,18,17,23,22,24);(2,6)(5,7)(11,19)(15,23)(18,24)(20,21)Polycyclic presentation:
2^8|a^2=g,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #109730, #109731, #116512, … |
| maximal normal subgroup of | #109730, #109731, #116512, … |
| maximal quotient of | #217268, #217269, #217270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14937);