Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 433 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,11,15,9,14)(7,12,18,26,19,27,24,31)(8,20,22,13,23,30,21,29)(17,32,33,28,34,36,35,25);(1,2,3,4)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,28)(14,31)(15,27)(16,26)(20,32)(21,35)(22,33)(23,34)(25,29)(30,36);(5,8,11,23)(6,13,15,29)(7,17,19,34)(9,22,10,21)(12,25,27,28)(14,30,16,20)(18,35,24,33)(26,36,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,e^b=eh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131347, #131349, #131475, … |
| maximal normal subgroup of | #131347, #131349, #131475, … |
| maximal quotient of | #115720, #115774, #115777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4062);