C₂×#6467359 #72224789

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#6467359

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁶
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

26825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226830, C₂×D₄×#3109
maximal normal subgroup of	#72226830, C₂×D₄×#3109
maximal quotient of	#72224795

GAP

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

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

C2×#6467359 #72224789