Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 16384 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,4,3)(5,6,12,17)(7,18,8,14)(9,23,26,36)(10,27,11,16)(13,22,29,20)(15,25,31,24)(19,33,21,28)(30,34,32,35);(1,3,4,2)(5,7,11,22,12,8,10,20)(6,13,27,14,17,29,16,18)(9,19,25,35,26,21,24,34)(15,28,36,30,31,33,23,32);(2,3)(5,8)(6,14)(7,12)(9,21)(10,22)(11,20)(13,16)(15,30)(17,18)(19,26)(23,28)(24,35)(25,34)(27,29)(31,32)(33,36);(5,9,11,25,12,26,10,24)(6,15,27,36,17,31,16,23)(7,19,22,35,8,21,20,34)(13,28,14,30,29,33,18,32)Polycyclic presentation:
2^8|a^2=fh,b^2=efh,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #502188, #502191, #502212, … |
| maximal normal subgroup of | #502188, #502191, #502212, … |
| maximal quotient of | #288032, #288037, #288042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18423);