#62988

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

5593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 20 points:

(2,4)(5,6,10,11)(7,12,17,20)(8,13,18,16)(9,14,19,15);(1,2,3,4);(6,11)(9,19)(12,20)(13,16);(5,7)(6,12)(8,14)(9,16)(10,17)(11,20)(13,19)(15,18);(5,8)(6,13)(7,15)(9,12)(10,18)(11,16)(14,17)(19,20);(5,9)(6,14)(7,16)(8,12)(10,19)(11,15)(13,17)(18,20)

Polycyclic presentation:

2^8|a^2=h,b^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	C₄⋊#5950, #11710797, #72341947, …
maximal subgroup of	#346522, #346530, #346538, …
maximal normal subgroup of	#346522, #346530, #346538, …
maximal quotient of	C₄⋊C₄×H(5,𝔽₂), #7625777, #7625796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55976);