Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = D4×C22⋊C4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,20)(3,13,16,8,17,26,21,25)(4,10,18,27,19,9,22,28)(14,30,15,23,32,24,31,29);(1,3,7,17)(2,8,12,25)(4,14,19,32)(5,21,6,16)(9,23,10,29)(11,26,20,13)(15,22,31,18)(24,27,30,28);(1,4)(2,9)(3,14)(5,18)(6,22)(7,19)(8,23)(10,12)(11,27)(13,24)(15,21)(16,31)(17,32)(20,28)(25,29)(26,30);(1,4)(2,10)(3,15)(5,22)(6,18)(7,19)(8,24)(9,12)(11,27)(13,23)(14,21)(16,32)(17,31)(20,28)(25,30)(26,29)Polycyclic presentation:
2^8|a^2=g,b^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #497324, #497326, #497330, … |
| maximal normal subgroup of | #497324, #497326, #497330, … |
| maximal quotient of | #220181, #220183, #220186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15447);