C₂×#500355 #72310719

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

Aliases

direct product

C₂×#500355

Subgroups

	subgroup	G/H
center, socle	C₂²	#24089
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#61267	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4925 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:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231596, C₂×#72213945
maximal normal subgroup of	C₂×#72231596, C₂×#72213945

GAP

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

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

C2×#500355 #72310719