Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 159 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:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,23,19,22,26,21,25)(20,24,28,27,29,31,30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,26,22,25,21,18)(20,27,28,31,29,32,30,24);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,28)(25,32)(26,31);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,25,26,19)(20,28,29,30)(24,27,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=ceh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #244482, #244483, #244484, … |
| maximal normal subgroup of | #244482, #244483, #244484, … |
| maximal quotient of | #355249, #355389, #355879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15292);