C₂×#492887 #72236745

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂×#492887

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#99
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁵
prime radical	C₂³×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄ ◁ C₂³×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

21641 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240260, #72256620, #72256634, …
maximal normal subgroup of	#72240260, #72256620, #72256634, …
maximal quotient of	#72236751

GAP

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

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

C2×#492887 #72236745