Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
588 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,16)(12,17)(13,22)(14,18)(15,19)(20,23)(21,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,22)(17,23)(18,24);(3,4)(6,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,24)(18,23);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,23,24,16)Polycyclic presentation:
2^8|d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244499, #244501, #244506, … |
| maximal normal subgroup of | #244499, #244501, #244506, … |
| maximal quotient of | #169617, #169635, #169682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11406);