Q₈×H(5,𝔽₂) #62982

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

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
prime radical	C₂×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

3703 conjugacy classes of subgroups; only the 3145 normal subgroups are shown.

normal subgroups only

This group has 3703 conjugacy classes of subgroups — too many to draw. Only its 3145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55970);