Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3330 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
303 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,5)(8,15)(9,13)(10,14)(11,16)(17,18,22,25)(19,23,21,30)(20,27,29,31)(24,28,32,26);(1,2)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,31,32,27)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^b=de,e^b=eg,f^c=fh,e^d=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20144, #6359483, #6359492, … |
| maximal normal subgroup of | C2×#20144, #6359483, #6359492, … |
| maximal quotient of | #222350, #222375, #222535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13132);