Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3506 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
533 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 533 conjugacy classes of subgroups — too many to draw. Only its 183 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:
(2,6)(3,4)(7,8)(9,10)(11,17,19,14)(12,16)(13,15)(18,23,24,21)(20,22);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23);(1,3,5,4)(2,6,7,8)(10,15)(12,20)(14,17)(18,24);(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=gh,e^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh49 conjugacy classes — view the full character table.
| maximal subgroup of | #221184, #221186, #221191, … |
| maximal normal subgroup of | #221184, #221186, #221191, … |
| maximal quotient of | #160644, #160645, #160650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10618);