Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 421 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,16)(7,12,20,29)(8,21,24,32)(9,15,25,14)(11,19,27,18)(13,30,33,23)(17,34,22,26)(28,36,31,35);(5,7,9,18,10,20,25,19)(6,11,14,12,16,27,15,29)(8,17,23,35,24,22,30,36)(13,26,21,28,33,34,32,31);(2,4)(5,7)(6,12)(8,22)(9,18)(10,20)(11,15)(13,31)(14,27)(16,29)(17,24)(19,25)(21,26)(23,36)(28,33)(30,35)(32,34);(5,8,9,23,10,24,25,30)(6,13,14,21,16,33,15,32)(7,17,18,35,20,22,19,36)(11,26,12,28,27,34,29,31)Polycyclic presentation:
2^8|a^2=fh,b^2=e,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501261, #503871, #510152, … |
| maximal normal subgroup of | #501261, #503871, #510152, … |
| maximal quotient of | #277356, #277358, #277361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17788);