C₂×#72286643 #72322851

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#72286643

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₈
Frattini, cosocle	C₈²	C₂⁵
socle	C₂³	C₂×#1580
prime radical	C₂×#33660	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂×#11402 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

4032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 50 points:

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

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72322903, #72323133

GAP

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

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

C2×#72286643 #72322851