Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
272 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 145 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) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,9,6,8,14,15)(3,7,11,4,10,12,16,13)(17,18)(19,27)(20,24)(21,33)(22,25)(23,29)(26,35)(28,41)(30,40)(31,39)(32,45)(34,37)(36,43)(38,44)(42,48)(46,47);(1,3)(2,7)(4,9)(5,11)(6,10)(8,12)(13,15)(14,16)(17,19,22,30)(18,23,25,37)(20,28,31,42)(21,34,35,29)(24,36,39,47)(26,40,33,27)(32,46,44,43)(38,48,45,41);(1,4,5,12,6,13,14,7)(2,3,9,11,8,10,15,16)(17,20,22,31)(18,24,25,39)(19,28,30,42)(21,32,35,44)(23,36,37,47)(26,38,33,45)(27,41,40,48)(29,43,34,46);(2,8)(3,10)(9,15)(11,16)(18,25)(20,31)(23,37)(26,33)(27,40)(28,42)(32,44)(43,46)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dgh,e^a=eg,d^b=dh,e^b=eg,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #136317, #136320, #137460, … |
| maximal normal subgroup of | #136317, #136320, #137460, … |
| maximal quotient of | #182488, #183646, #183915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10979);