Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,15,11,16,7,14)(8,12,9,13,17,20,18,19);(5,7,11,10)(6,12,16,20)(8,18,17,9)(13,15,19,14);(2,4)(5,8,11,17)(6,13,16,19)(7,9,10,18)(12,14,20,15)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,d^b=dh,e^b=egh,f^b=fgh,d^c=dg,e^c=eh,f^c=fh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| Fitting subgroup of | #12777⋊C3 |
| maximal subgroup of | #152558, #152586, #152587, … |
| maximal normal subgroup of | #152558, #152586, #152587, … |
| maximal quotient of | #129584, #129624, #130085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5765);