C₂×#4033842 #72296273

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#4033842

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

45787 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278026, C₂×#72277994, C₂×#72277950, …
maximal normal subgroup of	C₂×#72278026, C₂×#72277994, C₂×#72277950, …
maximal quotient of	C₂×#72264988, C₂×#72278125

GAP

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

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

C2×#4033842 #72296273