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) = C25 |
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 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 121 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)(4,8,15,21)(6,10,17,23)(12,26,27,20)(14,25,29,19)(16,22,30,31)(18,28,24,32);(1,3,5,13)(2,7,9,11)(4,12,15,27)(6,14,17,29)(8,20,21,26)(10,19,23,25)(16,28,30,32)(18,31,24,22);(1,4,6,16)(2,8,10,22)(3,12,14,28)(5,15,17,30)(7,18,19,20)(9,21,23,31)(11,24,25,26)(13,27,29,32);(4,15)(8,21)(12,27)(16,30)(18,24)(20,26)(22,31)(28,32)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^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 | #29657⋊C3 |
| maximal subgroup of | #245260, #245262, #245265, … |
| maximal normal subgroup of | #245260, #245262, #245265, … |
| maximal quotient of | #327787, #327788, #327789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22645);