C₂×#4308768 #72323939

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#4308768

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

134401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323247, C₂×#72232341, C₂×#72323255
maximal normal subgroup of	C₂×#72323247, C₂×#72232341, C₂×#72323255

GAP

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

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

C2×#4308768 #72323939