C₂×#72225212 #72231973

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#72225212

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#1368
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	C₂⁴	#1368
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

990 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);(3,4,5,9,6,10,11,16,7,8,12,14,13,15,17,18)(20,25,27,36,26,35,39,38)(21,29,30,43)(22,32)(23,34)(28,42,41,37,40,50,49,48)(31,46,45,44)(33,47);(4,8)(9,14)(10,15)(16,18)(20,26)(25,35)(27,39)(28,40)(36,38)(37,48)(41,49)(42,50);(19,20,21,25,22,28,29,37,23,27,31,38,32,41,44,42,24,26,30,35,33,40,43,48,34,39,45,36,47,49,46,50)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231993
maximal normal subgroup of	#72231993

GAP

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

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

C2×#72225212 #72231973