Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
222 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,17,19,24)(12,20,22,15)(13,16)(14,18)(21,23);(2,6)(3,4)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,d^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=cg,d^b=dh,e^b=efh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132198, #132200, #132202, … |
| maximal normal subgroup of | #132198, #132200, #132202, … |
| maximal quotient of | #125164, #125206, #125232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4552);