C₂×#498122 #72320664

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#498122

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33520	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 38 points:

(1,2);(3,4)(5,6)(7,8,12,18,31,24,11,22,14,9,20,23,13,19,15,17)(10,25,29,16,32,34,28,37,27,26,33,21,30,36,38,35);(7,9,11,22,13,24,12,18,14,8,15,17,31,19,20,23)(10,26,28,37,30,34,29,16,27,25,38,35,32,36,33,21);(7,10)(8,16,9,21)(11,29)(12,28)(13,32)(14,27)(15,33)(17,36,22,34)(18,26,23,25)(19,37,24,35)(20,38)(30,31);(4,6)(10,27)(16,21)(25,26)(28,38)(29,33)(30,32)(34,36)(35,37)

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72250872, C₂×#72250863
maximal normal subgroup of	C₂×#72250872, C₂×#72250863

GAP

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

gap> G := Group((1,2), (3,4)(5,6)(7,8,12,18,31,24,11,22,14,9,20,23,13,19,15,17)(10,25,29,16,32,34,28,37,27,26,33,21,30,36,38,35), (7,9,11,22,13,24,12,18,14,8,15,17,31,19,20,23)(10,26,28,37,30,34,29,16,27,25,38,35,32,36,33,21), (7,10)(8,16,9,21)(11,29)(12,28)(13,32)(14,27)(15,33)(17,36,22,34)(18,26,23,25)(19,37,24,35)(20,38)(30,31), (4,6)(10,27)(16,21)(25,26)(28,38)(29,33)(30,32)(34,36)(35,37));

C2×#498122 #72320664