Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 537 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) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,20)(15,22,23,18)(17,26,25,21)(19,27,28,24);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,23)(17,24)(19,21)(20,22)(25,27)(26,28);(2,4)(7,11)(10,12)(13,15)(14,19)(16,23)(17,27)(18,21)(20,28)(22,26)(24,25);(5,7,8,11)(6,10,9,12)(15,23)(18,22)(19,28)(24,27)Polycyclic presentation:
2^8|a^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29676, #6435010, #6435019, … |
| maximal normal subgroup of | C2×#29676, #6435010, #6435019, … |
| maximal quotient of | #328413, #328416, #328419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22664);