Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = C22×#120×#392 (order 8192) · Inn(G) = C24 · Out(G) = C2×#35680 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C24 |
| commutator, socle | C22 | C22×C42 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(2,6)(7,12)(8,13)(14,16)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(18,23)(22,28)(24,30)(29,32);(3,9)(7,12)(10,15)(14,16)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,f^2=h,b^a=bg,c^a=ch,d^c=dg88 conjugacy classes — view the full character table.
| maximal subgroup of | #128442, #184019, #184020, … |
| maximal normal subgroup of | #128442, #184019, #184020, … |
| maximal quotient of | #180807, #180826, #180853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10317);