C₂×#719696 #72298656

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂×#719696

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

82005 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233562, C₂×#72233569, C₂×#72233567, …
maximal normal subgroup of	C₂×#72233562, C₂×#72233569, C₂×#72233567, …
maximal quotient of	C₂×#72278211, C₂×#72250291, C₂×#72335144, …

GAP

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

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

C2×#719696 #72298656