C₂×#777542 #72325309

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#777542

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

59779 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)(15,16)(17,19)(18,20)(21,22);(3,4,5,6)(11,12)(13,14)(15,16,18,20)(17,19,22,21);(7,8,10,9)(11,12)(13,14);(4,6)(8,9)(11,12,13,14)(15,17,18,22)(16,19,20,21);(1,2);(3,4)(5,6)(7,9,10,8)(11,12)(13,14)(15,16)(17,21)(18,20)(19,22)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232187, C₂×#72232153, C₂×#72232166
maximal normal subgroup of	C₂×#72232187, C₂×#72232153, C₂×#72232166

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

C2×#777542 #72325309