C₂×#144639 #72258281

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂²	#10390
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	C₂×#3484	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3433 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,8)(5,11)(9,12,16,18)(13,14), (3,4,6,10)(5,12,13,16)(7,8,15,17)(9,14,18,11), (3,5,6,13)(4,9,10,18)(7,14,15,11)(8,16,17,12), (1,2));