Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
256 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 256 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:
(1,2)(3,6)(4,8)(5,7)(9,10,15,20)(11,21,12,23)(13,19,14,18)(16,24,17,22);(2,6)(3,5)(7,8)(9,11)(10,16)(12,15)(13,24)(14,22)(17,20)(18,23)(19,21);(1,2,4,8)(3,6,5,7)(9,12,15,11)(10,17,20,16)(13,24,14,22)(18,23,19,21)Polycyclic presentation:
2^8|a^2=gh,c^2=g,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dg,f^b=fh,d^c=dh,e^c=egh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10205, #237384, #237386, … |
| maximal normal subgroup of | C2×#10205, #237384, #237386, … |
| maximal quotient of | #102720, #102736, #102737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3193);