C₂×#36430 #7630110

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

12056 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

maximal subgroup of	C₂×#7684875, C₂×#7689177, C₂×#7689162, …
maximal normal subgroup of	C₂×#7684875, C₂×#7689177, C₂×#7689162, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537306);