Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,13)(12,14)(15,16)(19,26)(23,31)(27,32)(28,30);(1,2)(3,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,18)(19,27)(20,24)(21,25)(22,29)(23,28)(26,32)(30,31);(1,3,5,4)(2,6,7,8)(10,14)(13,16)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,31)(27,29);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,21,22)(18,24,25,29)(19,28,26,30)(23,32,31,27)Polycyclic presentation:
2^8|a^2=e,b^2=fh,c^2=e,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6633374, #6633379, #6633382, … |
| maximal normal subgroup of | #6633374, #6633379, #6633382, … |
| maximal quotient of | #383410, #384082, #384513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24579);