#60715

order	256
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#130035, #130044, #152088, …
maximal normal subgroup of	#130035, #130044, #152088, …
maximal quotient of	#6371189, #6372135, #6372456, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53703);