C₂×#656414 #72339413

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

Aliases

direct product

C₂×#656414

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

65489 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)(11,12,13,14)(15,16,18,20)(17,21,22,19);(15,16)(17,19)(18,20)(21,22);(8,10)(12,14)(15,16)(17,19)(18,20)(21,22);(1,2);(4,6)(7,8)(9,10)(11,12)(13,14)(15,17,18,22)(16,19,20,21);(3,4,5,6)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72257916, C₂×#72327612, C₂×#72327618
maximal normal subgroup of	C₂×#72257916, C₂×#72327612, C₂×#72327618

GAP

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

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

C2×#656414 #72339413