Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,26,29,23)(20,30,22,24)(27,32,28,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(19,27)(20,22)(23,31)(24,30)(26,32)(28,29);(3,7)(6,8)(17,20,21,22)(18,24,25,30)(19,28,29,27)(23,31,26,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29917, #6434093, #6434100, … |
| maximal normal subgroup of | C2×#29917, #6434093, #6434100, … |
| maximal quotient of | #379857, #380097, #380099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22905);