Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#396 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
480 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,5,10)(3,13,21,24,16,23,15,8)(4,9,18,25,20,26,19,27)(14,28,29,22,17,31,30,32);(1,3,5,15,6,16,7,21)(2,8,11,13,12,24,10,23)(4,17,19,29,20,14,18,30)(9,22,25,31,26,32,27,28);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,22)(10,25)(11,27)(12,26)(13,28)(15,29)(16,17)(21,30)(23,31)(24,32);(1,5,6,7)(2,10,12,11)(3,15,16,21)(4,18,20,19)(8,23,24,13)(9,25,26,27)(14,29,17,30)(22,31,32,28)Polycyclic presentation:
2^8|a^2=ef,b^2=e,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #23622⋊C3 |
| maximal subgroup of | #359387, #377686, #377936, … |
| maximal normal subgroup of | #359387, #377686, #377936, … |
| maximal quotient of | #242667, #242670, #242673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16610);