#26652

order	256
nilpotency class	3
isoclinism	isoclinic class of #23886
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

392 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)(3,6)(4,5)(7,8)(9,10,15,30,14,20,13,19)(11,21,29,16,25,36,24,32)(12,17,27,18,28,35,26,34)(22,37,23,31,39,33,38,40);(9,11)(10,16)(12,23)(13,29)(14,25)(15,24)(17,33)(18,31)(19,36)(20,32)(21,30)(22,27)(26,39)(28,38)(34,40)(35,37);(2,5)(6,8)(9,12)(10,17)(11,22)(13,26)(14,28)(15,27)(16,31)(18,30)(19,34)(20,35)(21,37)(23,29)(24,38)(25,39)(32,40)(33,36);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,27)(14,28)(15,26)(16,31)(17,30)(19,35)(20,34)(21,33)(22,29)(24,39)(25,38)(32,40)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#26652, #6439605, #6439606, …
maximal normal subgroup of	C₂×#26652, #6439605, #6439606, …
maximal quotient of	#310041, #310044, #310046, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 19640);