Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 147 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25,20,24,22,28)(19,23,27,31,26,30,29,32);(1,3,4,8)(2,5,6,7)(17,18,22,28,20,24,21,25)(19,23,29,32,26,30,27,31);(3,8)(5,7)(10,13)(14,16)(17,19)(18,23)(20,26)(21,27)(22,29)(24,30)(25,31)(28,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,26)(23,30)(27,29)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=f,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 | #140023, #140029, C2×#22338, … |
| maximal normal subgroup of | #140023, #140029, C2×#22338, … |
| maximal quotient of | #355304, #355671, #355804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15326);