#61198

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

Polycyclic presentation:

2^8|a^2=fh,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	#7003062, #7003064, #7003074, …
maximal normal subgroup of	#7003062, #7003064, #7003074, …
maximal quotient of	#6379087, #6379311, #6379721, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54186);