Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 546 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,11,13,18)(7,14)(12,19,20,15)(16,17);(1,2)(3,4)(5,6)(7,15)(8,18)(9,11)(10,13)(12,16)(14,19)(17,20);(2,4)(5,6)(7,12)(8,11)(9,18)(10,13)(14,20)(15,16)(17,19);(5,7)(6,12)(8,17)(9,16)(10,14)(11,19)(13,20)(15,18)Polycyclic presentation:
2^8|a^2=ef,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153129, #153130, #153282, … |
| maximal normal subgroup of | #153129, #153130, #153282, … |
| maximal quotient of | #343028, #343031, #343032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24061);