Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23)(25,26)(27,33,35,40)(28,32)(29,31)(30,34,38,36)(37,39);(2,6)(3,4)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,30)(28,36)(29,35)(31,38)(32,40)(33,39)(34,37);(1,3,5,4)(2,6,8,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,31)(30,38)(32,39)(33,40)Polycyclic presentation:
2^8|a^2=e,c^2=fh,d^2=g,e^2=h,b^a=bd,c^a=cfg,d^a=df,c^b=ch,d^b=dg,e^b=efg46 conjugacy classes — view the full character table.
| maximal subgroup of | #97308, #97309, #97310, … |
| maximal normal subgroup of | #97308, #97309, #97310, … |
| maximal quotient of | #96296, #96757, #96759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 714);