C₂³×#2537 #72289360

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

208149 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 18 points:

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

304 conjugacy classes. The character table is too large to display.

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

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