#62062

order	256
nilpotency class	3
isoclinism	isoclinic class of #61629
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 ◁ #529 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

902 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#62062, #10578870, #10578917, …
maximal normal subgroup of	C₂×#62062, #10578870, #10578917, …
maximal quotient of	#6386455, #6388582, #6388597, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55050);