#60713

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	#3058	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^8|b^2=h,c^2=g,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	#130033, #482714, #483077, …
maximal normal subgroup of	#130033, #482714, #483077, …
maximal quotient of	#6371188, #6372133, #6372134, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53701);