C₂×D₄×#124 #7629728

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

13810 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536924);