C₂³×#23900 #72291735

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²³

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁸ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3116000 conjugacy classes of subgroups.

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

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

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

368 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((1,2), (9,13), (7,8,10,12)(9,14)(11,13), (3,4)(5,6), (4,6)(7,9,10,13)(8,11,12,14), (15,16), (17,18));