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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
404 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(24,31)(25,26)(27,32);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,18,21,23)(19,27,29,32)(20,26,22,25)(24,30,31,28);(1,4,5,8)(2,6,3,7)(10,14)(13,15)(17,19,21,29)(18,24,23,31)(20,28,22,30)(25,27,26,32);(2,3)(6,7)(11,16)(13,15)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)Polycyclic presentation:
2^8|b^2=e,c^2=ef,d^2=h,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 | #6636821, #6636826, #6636866, … |
| maximal normal subgroup of | #6636821, #6636826, #6636866, … |
| maximal quotient of | #384283, #384289, #385765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25028);