C₂×C₅×#19745 #72249003

order	2560
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁷×C₈

Aliases

direct product

C₂×C₅×#19745

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁵×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

5996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

400 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	AGL(1,𝔽₁₁)×#19745
maximal quotient of	AGL(1,𝔽₁₁)×#19745

GAP

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

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

C2×C5×#19745 #72249003