Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,29)(21,31)(22,26)(23,27)(25,30)(28,32);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,27,22,28,30,20)(18,23,25,24,26,29,31,32);(11,15)(13,16)(17,19,21,27,22,28,30,20)(18,23,25,24,26,29,31,32);(1,3,4,8)(2,6,5,7)(17,20,30,28,22,27,21,19)(18,24,31,23,26,32,25,29)Polycyclic presentation:
2^8|a^2=efgh,b^2=f,c^2=f,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #504967, #504971, #504974, … |
| maximal normal subgroup of | #504967, #504971, #504974, … |
| maximal quotient of | #360927, #362674, #367306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21001);