Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
543 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 543 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,5,9)(3,11,13,7)(4,8,15,20)(6,10,17,22)(12,24,25,19)(14,23,27,18)(16,21,28,31)(26,30,32,29);(1,3,5,13)(2,7,9,11)(4,12,15,25)(6,14,17,27)(8,19,20,24)(10,18,22,23)(16,26,28,32)(21,29,31,30);(4,15)(7,18)(8,20)(11,23)(12,25)(16,28)(19,30)(21,31)(24,29)(26,32);(1,4,6,16)(2,8,10,21)(3,12,14,26)(5,15,17,28)(7,19,18,29)(9,20,22,31)(11,24,23,30)(13,25,27,32)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,d^c=de37 conjugacy classes — view the full character table.
| Fitting subgroup of | #29658⋊C3 |
| maximal subgroup of | #246383, #246384, #246387, … |
| maximal normal subgroup of | #246383, #246384, #246387, … |
| maximal quotient of | #327803, #327804, #327805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22646);