C₂×#245711 #72233859

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂×#245711

Subgroups

	subgroup	G/H
center, socle	C₂³	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

13317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213883, #72240334, #72257944, …
maximal normal subgroup of	C₂×#72213883, #72240334, #72257944, …
maximal quotient of	#72261567, #72264471, #72264682, …

GAP

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

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

C2×#245711 #72233859