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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 474 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:
(6,7)(10,11)(13,14,17,21)(15,18,22,26)(16,19,24,20)(23,25,28,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,22)(14,18,21,26)(16,23,24,28)(19,25,20,27);(2,4)(5,7,9,6)(8,10,12,11)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(5,8,9,12)(6,10,7,11)(15,22)(18,26)(23,28)(25,27)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=e,d^2=e,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 | #269171, #269173, #269175, … |
| maximal normal subgroup of | #269171, #269173, #269175, … |
| maximal quotient of | #329029, #329030, #329032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22687);