Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3460 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(19,25)(22,28)(26,31)(29,32);(3,10)(7,14)(11,16)(13,15)(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^8|a^2=f,e^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=ch,e^b=eg,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17505, #6343015, #6343025, … |
| maximal normal subgroup of | C2×#17505, #6343015, #6343025, … |
| maximal quotient of | #162948, #162950, #162952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10493);