Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C2×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
883 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 883 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,10)(8,11)(9,12)(13,18)(14,17)(15,19)(16,20);(1,2)(3,4)(5,7)(6,10)(8,13)(9,14)(11,18)(12,17)(15,20)(16,19);(1,3,4,2)(5,8)(6,11)(7,13)(9,15)(10,16)(12,19)(14,20)(17,18);(8,15)(11,19)(13,20)(16,18)Polycyclic presentation:
2^8|c^2=e,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #394547, #394548, #394724, … |
| maximal normal subgroup of | #394547, #394548, #394724, … |
| maximal quotient of | #238450, #238451, #238452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16503);