C₂×#1421062 #72331779

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

Aliases

direct product

C₂×#1421062

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(8,10)(11,12)(13,16)(14,17)(15,18);(4,6)(7,8)(9,10)(11,12,14,17)(13,15,18,16)(19,20,23,25)(21,22,24,26);(11,13)(12,15)(14,18)(16,17)(19,20,23,25)(21,26,24,22);(1,2);(3,4,5,6)(11,13)(12,16)(14,18)(15,17)(19,21)(20,22)(23,24)(25,26);(11,12)(13,15)(14,17)(16,18)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323334, C₂×#72323308, C₂×#72323316
maximal normal subgroup of	C₂×#72323334, C₂×#72323308, C₂×#72323316

GAP

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

gap> G := Group((8,10)(11,12)(13,16)(14,17)(15,18), (4,6)(7,8)(9,10)(11,12,14,17)(13,15,18,16)(19,20,23,25)(21,22,24,26), (11,13)(12,15)(14,18)(16,17)(19,20,23,25)(21,26,24,22), (1,2), (3,4,5,6)(11,13)(12,16)(14,18)(15,17)(19,21)(20,22)(23,24)(25,26), (11,12)(13,15)(14,17)(16,18)(19,22)(20,24)(21,25)(23,26));

C2×#1421062 #72331779