#62314

order	256
nilpotency class	3
isoclinism	isoclinic class of #62307
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

897 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24)(13,18)(14,28)(15,19)(17,26)(20,35)(21,30)(22,31)(23,32)(25,34)(27,33)(29,37)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,23)(10,16,19,32)(12,20,27,38)(13,21,14,22)(17,29,34,39)(18,30,28,31)(24,35,33,40)(25,36,26,37);(9,12,14,26,15,27,13,25)(10,17,18,24,19,34,28,33)(11,20,22,37,23,38,21,36)(16,29,30,35,32,39,31,40);(3,8)(5,7)(9,13,15,14)(10,18,19,28)(11,21,23,22)(12,25,27,26)(16,30,32,31)(17,24,34,33)(20,36,38,37)(29,35,39,40);(2,6)(5,7)(9,13,15,14)(10,18,19,28)(11,22,23,21)(12,26,27,25)(16,31,32,30)(17,33,34,24)(20,36,38,37)(29,35,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#62314, #10577613, #10577659, …
maximal normal subgroup of	C₂×#62314, #10577613, #10577659, …
maximal quotient of	#6386800, #6389160, #6389218, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55302);