D₄×#557 #7699501

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

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

7101 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606697);