C₂×#94176 #72340586

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#94176

Subgroups

	subgroup	G/H
center, socle	C₂³	#1393
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	#2297	C₂³
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂×#2297 ◁ G
LECS	G ▷ #2297 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340597, #72340605, C₂×#72213278
maximal normal subgroup of	#72340597, #72340605, C₂×#72213278

GAP

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

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

C2×#94176 #72340586