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) = C24×C4 |
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 | #3496 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
635 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 635 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:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(2,5)(3,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(2,6)(5,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(12,21)(15,20)(18,24)(22,23)Polycyclic presentation:
2^8|a^2=f,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 | #253507, #253513, #253519, … |
| maximal normal subgroup of | #253507, #253513, #253519, … |
| maximal quotient of | #160630, #160631, #160892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10608);