C₂×#497864 #72285523

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

Aliases

direct product

C₂×#497864

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4043 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×AGL(1,ℤ/16), #72310001, #72310080, …
maximal normal subgroup of	C₂×D₄×AGL(1,ℤ/16), #72310001, #72310080, …
maximal quotient of	C₂×#72226978, C₂×#72250110, #72310001, …

GAP

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

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

C2×#497864 #72285523