C₂×#1082756 #72331811

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#1082756

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

65987 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323324, C₂×#72323308, C₂×#72323304
maximal normal subgroup of	C₂×#72323324, C₂×#72323308, C₂×#72323304

GAP

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

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

C2×#1082756 #72331811