C₂×#484073 #72322211

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#484073

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₈
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂³	C₂×C₄×D₈
prime radical	C₂³×D₁₆	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ C₂×C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

(12,16)(13,19)(15,21)(17,24)(18,23)(20,25)(22,26);(3,4,7,6)(5,9,10,8);(3,5,6,9,7,10,4,8)(11,12,14,16)(13,20,22,17)(15,18,23,21)(19,24,26,25);(1,2);(11,13,21,24,16,20,23,26,14,22,18,25,12,17,15,19)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232027, #72323054, #72323072, …
maximal normal subgroup of	C₂×#72232027, #72323054, #72323072, …
maximal quotient of	#72322334, #72322501

GAP

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

gap> G := Group((12,16)(13,19)(15,21)(17,24)(18,23)(20,25)(22,26), (3,4,7,6)(5,9,10,8), (3,5,6,9,7,10,4,8)(11,12,14,16)(13,20,22,17)(15,18,23,21)(19,24,26,25), (1,2), (11,13,21,24,16,20,23,26,14,22,18,25,12,17,15,19));

C2×#484073 #72322211