#62986

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

3947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 16 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14);(9,11,13,16)(10,14,12,15);(2,5)(6,8);(3,7)(6,8)(9,10,13,12)(11,14,16,15);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14);(1,2)(3,6)(4,5)(7,8)

Polycyclic presentation:

2^8|a^2=g,b^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,f^c=fh,e^d=eh

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710791, #11710792, #11710793
maximal subgroup of	#347228, #347252, #347276, …
maximal normal subgroup of	#347228, #347252, #347276, …
maximal quotient of	#7625776, #7625801, #7625804, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55974);