Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
381 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 256 |
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)(21,22,24,28,25,27,33,35)(23,29,30,34,31,36,32,26);(1,2,3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19);(2,4)(8,16)(11,18)(14,20)(17,19)(21,23,25,31)(22,26,27,34)(24,32,33,30)(28,36,35,29);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,27)(26,34)(28,35)(29,36)Polycyclic presentation:
2^8|a^2=ef,b^2=eg,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #102693, #102694, #102696, … |
| maximal normal subgroup of | #102693, #102694, #102696, … |
| maximal quotient of | #331536, #331540, #331544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23272);