C₂×#5955312 #72331091

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

Aliases

direct product

C₂×#5955312

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#534	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

71325 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323284, C₂×#72323288, C₂×#72323282
maximal normal subgroup of	C₂×#72323284, C₂×#72323288, C₂×#72323282

GAP

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

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

C2×#5955312 #72331091