C₂×#126708 #72319086

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#126708

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#106
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂⁵	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 38 points:

(3,4,5,6)(7,8)(9,12)(10,14)(11,13)(15,19)(16,18)(17,20)(21,22)(23,24,25,28,26,30,31,35,27,29,32,34,33,36,38,37);(1,2);(7,9,10,15,11,16,17,21)(8,12,14,19,13,18,20,22)(24,28,29,34)(25,31)(26,33)(30,37,36,35)(32,38);(8,13)(12,18)(14,20)(19,22)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224980, C₂×#72225109, C₂×#72225106, …
maximal normal subgroup of	C₂×#72224980, C₂×#72225109, C₂×#72225106, …
maximal quotient of	C₂×#72224978

GAP

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

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

C2×#126708 #72319086