Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 103 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 — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,21,19)(20,25,28,23)(22,24,30,26)(27,31,32,29);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,18)(20,27,28,32)(22,24,30,26)(23,29,25,31);(3,7)(6,8)(10,14)(13,16)(17,20,21,28)(18,23,19,25)(22,31,30,29)(24,27,26,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,19)(23,25)(24,26)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #32805⋊C3 |
| maximal subgroup of | C2×#32805, #6631709, #6631716, … |
| maximal normal subgroup of | C2×#32805, #6631709, #6631716, … |
| maximal quotient of | #391270, #391271, #391297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25793);