C₂²×D₄×Q₈ #62696

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

C₂²×D₄×Q₈

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

5899 conjugacy classes of subgroups; only the 3479 normal subgroups are shown.

normal subgroups only

This group has 5899 conjugacy classes of subgroups — too many to draw. Only its 3479 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) = 128

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55684);