#26650

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	#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

391 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)(9,10,13,18,14,19,29,33)(11,20,23,31,24,36,28,15)(12,16,25,34,27,35,26,17)(21,37,38,40,39,32,22,30);(9,11,14,24)(10,15,19,31)(12,22,27,38)(13,28,29,23)(16,32,35,37)(17,30,34,40)(18,36,33,20)(21,25,39,26);(2,5)(6,8)(9,12)(10,16)(11,21)(13,25)(14,27)(15,30)(17,33)(18,34)(19,35)(20,37)(22,28)(23,38)(24,39)(26,29)(31,40)(32,36);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,30)(16,33)(18,35)(19,34)(20,32)(21,28)(23,39)(24,38)(25,29)(31,40)(36,37)

Polycyclic presentation:

2^8|a^2=e,b^2=h,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×#26650, #6439605, #6439607, …
maximal normal subgroup of	C₂×#26650, #6439605, #6439607, …
maximal quotient of	#309993, #309996, #309998, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 19638);