C₂×#509107 #72237293

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

Aliases

direct product

C₂×#509107

Subgroups

	subgroup	G/H
center, socle	C₂²	#30396
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4293 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);(4,9,13,25)(6,20)(10,22,21,24)(11,27)(12,29,15,23)(17,32)(26,34,30,33)(28,31);(4,10,13,21)(5,14)(6,11)(7,16)(9,22,25,24)(12,30,15,26)(17,31)(20,27)(23,34,29,33)(28,32);(3,4,11,22,16,25,6,21,8,13,27,24,7,9,20,10)(5,15,28,34,18,23,17,26,14,12,31,33,19,29,32,30);(3,5,16,18,8,14,7,19)(4,12,25,29,13,15,9,23)(6,17,27,31,20,32,11,28)(10,26,22,33,21,30,24,34)

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237280, #72250127, #72250414, …
maximal normal subgroup of	C₂×#72237280, #72250127, #72250414, …
maximal quotient of	#72250127, #72250414, #72322998

GAP

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

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

C2×#509107 #72237293