C₂×#7768 #165738

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1083 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4,6,9)(5,7,10,8)(15,18)(20,23)(21,25)(24,26);(4,7)(5,10)(8,9)(11,12,14,16)(13,15,17,18);(11,13,14,17)(12,15,16,18)(19,20,22,24)(21,26,25,23)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72934);