#26851

order	256
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,23)(12,29)(13,33)(14,21)(15,20)(16,22)(17,26)(18,31)(19,24)(25,39)(27,36)(28,35)(30,37)(32,38)(34,40);(1,3,5,4)(2,6,7,8)(9,11,15,27,16,28,14,26)(10,17,21,35,22,36,20,23)(12,30,31,24,32,13,25,34)(18,37,29,40,39,33,38,19);(1,4,5,3)(2,6,7,8)(9,12,14,25,16,32,15,31)(10,18,20,38,22,39,21,29)(11,24,26,30,28,34,27,13)(17,19,23,33,36,40,35,37);(2,7)(6,8)(9,13,14,24,16,30,15,34)(10,19,20,33,22,40,21,37)(11,25,26,32,28,31,27,12)(17,18,23,38,36,39,35,29)

Polycyclic presentation:

2^8|b^2=eh,c^2=e,d^2=fh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,g^c=gh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#26851, #6443957, #6443964, …
maximal normal subgroup of	C₂×#26851, #6443957, #6443964, …
maximal quotient of	#311010, #311011, #311013, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 19839);