C₂×#4033849 #72325577

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#4033849

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

60097 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232169, C₂×#72232083, C₂×#72232101, …
maximal normal subgroup of	C₂×#72232169, C₂×#72232083, C₂×#72232101, …

GAP

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

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

C2×#4033849 #72325577