Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
958 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 958 conjugacy classes of subgroups — too many to draw. Only its 181 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)(3,12)(4,17)(5,22)(6,11)(7,10)(8,14)(9,20)(13,30)(15,24)(16,25)(18,29)(19,27)(21,28)(23,31)(26,32);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,22)(13,19)(17,23)(20,31)(21,32)(27,29)(28,30);(1,4)(2,9)(3,13)(5,19)(6,20)(7,21)(8,23)(10,27)(11,28)(12,29)(14,18)(15,32)(16,31)(17,22)(24,30)(25,26);(2,10)(8,24)(9,27)(11,22)(12,25)(17,28)(23,30)(26,29)Polycyclic presentation:
2^8|e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #23434⋊C3 |
| maximal subgroup of | #142262, #142263, #393928, … |
| maximal normal subgroup of | #142262, #142263, #393928, … |
| maximal quotient of | #232445, #232446, #232447, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16422);