C₂×#33106 #6629653

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#2856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3354 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

maximal subgroup of	C₂×#6630193, C₂×#6578604, C₂×D₄×#480, …
maximal normal subgroup of	C₂×#6630193, C₂×#6578604, C₂×D₄×#480, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536849);