Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C2×#120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
561 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 113 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)(11,15)(12,14)(13,16)(17,20)(18,23)(19,21)(22,24);(1,2)(3,6)(4,8)(5,7)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(3,4)(6,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(1,3,5,4)(2,6,7,8)(10,16)(11,18)(14,17)(19,24)Polycyclic presentation:
2^8|d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #250775, #250777, #250804, … |
| maximal normal subgroup of | #250775, #250777, #250804, … |
| maximal quotient of | #167037, #167055, #167102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11260);