Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
425 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,29)(20,33)(21,28)(22,27)(23,26)(24,31)(25,34)(30,38)(32,41)(35,42)(36,39)(37,40)(43,45)(44,47)(46,48);(4,13)(7,12)(9,16)(11,14)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,39)(27,41)(28,29)(33,47)(34,45)(35,46)(37,44)(40,48)(42,43);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,20,23,35)(18,25,26,40)(19,30,32,46)(21,34,22,37)(24,38,39,48)(27,33,28,42)(29,43,41,47)(31,44,36,45);(18,26)(24,39)(25,40)(27,28)(29,41)(33,42)(38,48)(43,47)Polycyclic presentation:
2^8|a^2=h,c^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153202, #153204, #245960, … |
| maximal normal subgroup of | #153202, #153204, #245960, … |
| maximal quotient of | #165880, #165900, #165913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11987);