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) = C23 |
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 | |
175 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,12,15,14,18,11,19,13,16,20,23,21,24,22,17);(1,2)(3,6)(4,8)(5,7)(10,15)(11,20)(12,22)(14,21)(16,23)(17,18)(19,24);(1,3,5,4)(2,6,7,8)(10,16)(15,23)(17,19)(18,24)Polycyclic presentation:
2^8|a^2=e,c^2=fh,d^2=gh,e^2=fgh,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 | #130431, #130433, #130448, … |
| maximal normal subgroup of | #130431, #130433, #130448, … |
| maximal quotient of | #124297, #124423, #124731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4970);