C₂×#210752 #72321176

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#210752

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂×C₄×D₄
prime radical	C₂²×#466	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72318326, C₂×#72318344, C₂×#72318337, …
maximal normal subgroup of	C₂×#72318326, C₂×#72318344, C₂×#72318337, …

GAP

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

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

C2×#210752 #72321176