Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| 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, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 425 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)(5,6,9,14,10,13,22,28)(7,11,17,25,16,24,32,35)(8,18,19,29,20,12,21,27)(15,23,31,34,30,26,33,36);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,26)(13,24)(14,25)(18,23)(19,31)(20,30)(21,33)(22,32)(27,36)(28,35)(29,34);(5,8,10,20)(6,12,13,18)(7,15,16,30)(9,21,22,19)(11,23,24,26)(14,29,28,27)(17,33,32,31)(25,36,35,34)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131339, #131341, #131467, … |
| maximal normal subgroup of | #131339, #131341, #131467, … |
| maximal quotient of | #115702, #115726, #115727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4036);