C₂×#2601318 #72323935

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#2601318

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

51515 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323247, C₂×#72232176, C₂×#72297911, …
maximal normal subgroup of	C₂×#72323247, C₂×#72232176, C₂×#72297911, …

GAP

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

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

C2×#2601318 #72323935