Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,10,7)(8,11,9,12)(13,14,19,16)(15,22,25,28)(17,27,18,21)(20,24,26,23);(6,11)(7,12)(8,9)(13,15,19,25)(14,20,16,26)(17,24,18,23)(21,28,27,22);(2,4)(5,7,10,6)(8,12,9,11)(13,16,19,14)(15,22,25,28)(17,27,18,21)(20,23,26,24)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #93218, #125880, #125882, … |
| maximal normal subgroup of | #93218, #125880, #125882, … |
| maximal quotient of | #102440, #102456, #102468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2961);