C₂×#500351 #72238313

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#500351

Subgroups

	subgroup	G/H
center, socle	C₂²	#24089
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#61259	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(5,13,15,26)(6,17)(7,19)(10,22,14,25)(11,24)(12,21)(16,30,29,27)(18,31)(20,32)(23,34,28,33);(3,4)(5,14)(6,11)(7,12)(8,9)(10,15)(13,25)(16,28)(17,24)(18,20)(19,21)(22,26)(23,29)(27,34)(30,33)(31,32);(4,9)(11,20)(12,21)(24,32);(3,5,6,13,7,16,17,27,8,15,18,26,19,29,31,30)(4,10,11,22,12,23,24,33,9,14,20,25,21,28,32,34)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213937, #72250731, #72252912, …
maximal normal subgroup of	C₂×#72213937, #72250731, #72252912, …
maximal quotient of	#72252912, #72257102, #72307231, …

GAP

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

gap> G := Group((1,2), (5,13,15,26)(6,17)(7,19)(10,22,14,25)(11,24)(12,21)(16,30,29,27)(18,31)(20,32)(23,34,28,33), (3,4)(5,14)(6,11)(7,12)(8,9)(10,15)(13,25)(16,28)(17,24)(18,20)(19,21)(22,26)(23,29)(27,34)(30,33)(31,32), (4,9)(11,20)(12,21)(24,32), (3,5,6,13,7,16,17,27,8,15,18,26,19,29,31,30)(4,10,11,22,12,23,24,33,9,14,20,25,21,28,32,34));

C2×#500351 #72238313