C₂×#742738 #72302713

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#742738

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2397	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

25345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280169, C₂×#72280294, C₂×#72280470
maximal normal subgroup of	C₂×#72280169, C₂×#72280294, C₂×#72280470

GAP

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

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

C2×#742738 #72302713