C₂×#647681 #72341749

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#647681

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

58033 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72268528, C₂×#72303265, C₂×#72268529
maximal normal subgroup of	C₂×#72268528, C₂×#72303265, C₂×#72268529

GAP

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

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

C2×#647681 #72341749