Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
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 | #3169 | C2 |
| 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 | |
170 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,19,24,27,31,32,29,21,23,28,30,20,25,26,22);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,22)(19,26)(20,27)(23,29)(24,25)(28,32)(30,31);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(18,23)(22,29)(24,30)(25,31)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=gh,e^2=fgh,g^2=h,b^a=bd,c^a=ch,d^a=df,d^b=dg,e^b=efg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129015, #129033, #131636, … |
| maximal normal subgroup of | #129015, #129033, #131636, … |
| maximal quotient of | #124109, #124182, #124968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4939);