C₂×#29734 #6433018

order	512
nilpotency class	3
isoclinism	isoclinic class of #29730
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#2847	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3888 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

maximal subgroup of	C₂×#6724588, C₂×#6677836, C₂×#6720377, …
maximal normal subgroup of	C₂×#6724588, C₂×#6677836, C₂×#6720377, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340214);