#60705

order	256
nilpotency class	3
isoclinism	isoclinic class of #60454
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	#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

923 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)(9,10,14,18)(11,15,20,28)(12,22,23,37)(13,25,26,17)(16,30,31,24)(19,32,33,40)(21,36,35,29)(27,38,39,34);(2,5)(6,8)(9,11,14,20)(10,15,18,28)(12,19,23,33)(13,21,26,35)(16,27,31,39)(17,29,25,36)(22,32,37,40)(24,34,30,38);(9,12,13,24,14,23,26,30)(10,16,17,22,18,31,25,37)(11,19,21,34,20,33,35,38)(15,27,29,32,28,39,36,40);(1,3,4,7)(2,6,5,8)(11,20)(15,28)(19,33)(21,35)(27,39)(29,36)(32,40)(34,38);(12,23)(16,31)(19,33)(22,37)(24,30)(27,39)(32,40)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#152081, #152090, #246935, …
maximal normal subgroup of	#152081, #152090, #246935, …
maximal quotient of	#6371186, #6372125, #6372444, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53693);