Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1441 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,11,5,22)(3,12,15,25,16,24,14,8)(4,17,20,28,21,27,19,9)(13,23,29,26,30,31,18,32);(1,3,5,14,7,16,6,15)(2,8,10,12,11,25,22,24)(4,18,19,30,21,29,20,13)(9,26,17,31,28,32,27,23);(1,4,5,19,7,21,6,20)(2,9,10,17,11,28,22,27)(3,13,14,18,16,30,15,29)(8,23,12,26,25,31,24,32)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11254, #359161, #360230, … |
| maximal normal subgroup of | C2×#11254, #359161, #360230, … |
| maximal quotient of | #120178, #120206, #120207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4242);