Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| 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 | C22×C4⋊C4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(19,25)(22,28)(26,31)(29,32);(3,10)(8,15)(11,14)(12,16)Polycyclic presentation:
2^8|a^2=f,b^2=eg,e^2=g,f^2=h,b^a=be,e^a=eg,c^b=ch,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #225506, #225507, #225508, … |
| maximal normal subgroup of | #225506, #225507, #225508, … |
| maximal quotient of | #180925, #183374, #183578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10800);