Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 365 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,18,12,15)(11,19,22,24,20,23,21,16);(1,3,4,7)(2,5,6,8)(10,15,18,17)(11,20)(16,19,24,23)(21,22);(2,6)(5,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,21)(20,24)(22,23);(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23)Polycyclic presentation:
2^8|a^2=gh,b^2=ef,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #504677, #505193, #506180, … |
| maximal normal subgroup of | #504677, #505193, #506180, … |
| maximal quotient of | #315086, #315089, #315093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20799);