C₂×#212949 #72321551

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#212949

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×#493	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5865 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)(7,8,11,13,12,14,10,9)(16,19)(17,20)(21,22);(1,2);(7,8,11,13,12,14,10,9)(15,16,20,22,18,21,17,19);(7,9,10,14,12,13,11,8)(16,21)(19,22);(3,4,5,6)(7,10,12,11)(8,13,14,9)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72318344, C₂×#72318337, C₂×#72318392, …
maximal normal subgroup of	C₂×#72318344, C₂×#72318337, C₂×#72318392, …

GAP

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

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

C2×#212949 #72321551