C₂⁴×#2408 #72291072

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁹

Aliases

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10211752 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 20 points:

(17,18,20,19);(3,4);(10,12)(14,16)(18,19);(14,16)(17,19,20,18);(5,6);(9,10)(11,12)(13,14)(15,16);(7,8);(1,2)(3,4)(5,6)(7,8)

608 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((17,18,20,19), (3,4), (10,12)(14,16)(18,19), (14,16)(17,19,20,18), (5,6), (9,10)(11,12)(13,14)(15,16), (7,8), (1,2)(3,4)(5,6)(7,8));