Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 115 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,28,32,31);(2,5)(6,8)(10,13)(14,15)(17,19,21,27)(18,23,22,25)(20,28,29,31)(24,26,30,32);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,29)(18,24,22,30)(19,26,27,32)(23,31,25,28);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(20,29)(24,30)(26,32)(28,31)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=efh,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #100564, #100566, #168836, … |
| maximal normal subgroup of | #100564, #100566, #168836, … |
| maximal quotient of | #380415, #380443, #381243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23104);