Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C23 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 63 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 — 3 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,22,26,24,25,30,32,23,27,21,31,20,19,28,29);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,25)(18,24,27,20)(21,31,22,26)(28,32,30,29);(2,7)(6,9)(8,15)(14,16)(17,20,23,24)(18,25,27,19)(21,30,22,28)(26,32,31,29)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=gh,e^2=fg,g^2=h,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131096, #131097, #131100, … |
| maximal normal subgroup of | #131096, #131097, #131100, … |
| maximal quotient of | #124274, #124471, #124472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4956);