Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | #532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10,14,20,16,19,28,36,15,21,27,37,13,29,12,18)(11,22,25,32,24,17,31,33,26,35,39,40,30,34,23,38);(2,5)(3,7)(9,11)(10,17)(12,23)(13,30)(14,25)(15,26)(16,24)(18,32)(19,35)(20,33)(21,34)(22,29)(27,39)(28,31)(36,40)(37,38);(9,12,13,27,15,28,16,14)(10,18,29,37,21,36,19,20)(11,23,30,39,26,31,24,25)(17,32,22,38,34,40,35,33)Polycyclic presentation:
2^8|a^2=e,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,d^b=dh,e^b=efh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94149, #94150, #128504, … |
| maximal normal subgroup of | #94149, #94150, #128504, … |
| maximal quotient of | #124201, #124210, #124237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4478);