#61197

order	256
nilpotency class	3
isoclinism	isoclinic class of #61071
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
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #525 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1023 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#7003063, #7003065, #7003073, …
maximal normal subgroup of	#7003063, #7003065, #7003073, …
maximal quotient of	#6379087, #6379312, #6379722, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54185);