Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,26,34,25,30,24,29)(23,27,31,28,33,36,32,35);(1,2,3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,27)(24,32)(25,33)(26,31)(28,34)(29,35)(30,36);(2,4)(8,16)(11,18)(14,20)(17,19)(21,23)(22,28)(24,31)(25,33)(26,32)(27,34)(29,36)(30,35);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,25,26)(22,29,30,34)(23,31,33,32)(27,28,36,35)Polycyclic presentation:
2^8|a^2=efh,b^2=eg,d^2=eh,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #300485, #300486, #300487, … |
| maximal normal subgroup of | #300485, #300486, #300487, … |
| maximal quotient of | #331538, #331542, #331545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23278);