SD₁₆×C₄⋊Q₈ #6583234

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2060 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

maximal subgroup of	#72285147, #72285207
maximal normal subgroup of	#72285147, #72285207

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490430);