Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11757 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
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 | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(2,6)(3,4)(7,8)(9,10,13,18,14,17,11,15)(12,20,21,23,22,16,19,24);(1,2,5,8)(3,6,4,7)(9,11,14,13)(10,15)(12,21,22,19)(16,23)(17,18)(20,24);(1,3,5,4)(2,7,8,6)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)Polycyclic presentation:
2^8|a^2=e,b^2=fh,c^2=gh,d^2=gh,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cfh,d^b=dg,e^b=efgh,d^c=dh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #130298, #130300, #130301, … |
| maximal normal subgroup of | #130298, #130300, #130301, … |
| maximal quotient of | #124312, #124322, #124486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4896);