Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
533 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 533 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(9,10,13,11)(12,15,16,14)(17,18,21,20)(19,23,24,22)(27,31)(29,32);(2,5)(6,8)(10,11)(14,15)(19,24)(22,23)(25,26,28,30)(27,29,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,22,20,23)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,23,24,22)(25,27)(26,29)(28,31)(30,32)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,c^2=ef,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #104745, #104746, #104747, … |
| maximal normal subgroup of | #104745, #104746, #104747, … |
| maximal quotient of | #154191, #154409, #154443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8243);