Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(6,7)(9,10,12,13)(11,14,16,15)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30);(1,2,5,3)(4,6,8,7)(9,11,12,16)(10,14,13,15)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30);(1,3,5,2)(4,6,8,7)(10,13)(11,15)(14,16)(20,29)(23,32)(26,28)(30,31);(1,4,5,8)(2,6,3,7)(11,16)(14,15)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,c^2=fh,d^2=e,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #102536, #102537, #102540, … |
| maximal normal subgroup of | #102536, #102537, #102540, … |
| maximal quotient of | #391390, #391428, #391490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25838);