C₂×#72225221 #72323094

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

Aliases

direct product

C₂×#72225221

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×#33733	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×#1312 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

3992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

#72323108, #72323164

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

C2×#72225221 #72323094