Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,30)(21,25)(22,24)(23,28)(27,31)(29,32);(1,2,4,5)(3,6,7,8)(17,19,20,27,21,28,22,29)(18,23,30,32,25,26,24,31);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,21,28)(18,23,25,26)(20,29,22,27)(24,32,30,31);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,24,25,30)(19,27,28,29)(23,31,26,32)Polycyclic presentation:
2^8|b^2=eg,c^2=h,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30370, #6433924, #6433933, … |
| maximal normal subgroup of | C2×#30370, #6433924, #6433933, … |
| maximal quotient of | #380919, #380923, #380944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23358);