C₂×#72225141 #72231963

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#72225141

Subgroups

	subgroup	G/H
center	C₂×C₄²	#322
commutator	C₈	C₂²×C₄×C₁₆
Frattini, cosocle	#1311	C₂⁴
socle	C₂³	C₂×#1313
prime radical	C₂⁴	#1313
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(3,4,5,8,6,9,11,14,7,10,12,15,13,16,17,18)(20,25,27,37,26,36,29,39)(21,30,31,41)(22,33)(28,43,44,49,40,50,42,38)(32,47,46,45)(34,48);(20,26)(25,36)(27,29)(28,40)(37,39)(38,49)(42,44)(43,50);(19,20,21,28,22,25,32,38,23,29,30,44,34,39,45,43,24,26,31,40,33,36,46,49,35,27,41,42,48,37,47,50)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72231972
maximal normal subgroup of	#72231972

GAP

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

gap> G := Group((1,2), (3,4,5,8,6,9,11,14,7,10,12,15,13,16,17,18)(20,25,27,37,26,36,29,39)(21,30,31,41)(22,33)(28,43,44,49,40,50,42,38)(32,47,46,45)(34,48), (20,26)(25,36)(27,29)(28,40)(37,39)(38,49)(42,44)(43,50), (19,20,21,28,22,25,32,38,23,29,30,44,34,39,45,43,24,26,31,40,33,36,46,49,35,27,41,42,48,37,47,50));

C2×#72225141 #72231963