Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
604 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 604 conjugacy classes of subgroups — too many to draw. Only its 173 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 — 3 generators on 32 points:
(1,2,6,10,7,11,5,19)(3,12,16,22,15,27,14,8)(4,9,17,24,18,25,20,23)(13,28,29,21,30,26,31,32);(1,3)(2,8,11,22)(4,13)(5,14)(6,16)(7,15)(9,21,25,32)(10,12,19,27)(17,29)(18,30)(20,31)(23,28,24,26);(1,4)(2,9)(3,13)(5,20)(6,17)(7,18)(8,21)(10,24)(11,25)(12,26)(14,31)(15,30)(16,29)(19,23)(22,32)(27,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146595, #146615, #146650, … |
| maximal normal subgroup of | #146595, #146615, #146650, … |
| maximal quotient of | #106089, #106090, #106098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3731);