C₂×#498377 #72320624

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

Aliases

direct product

C₂×#498377

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33605	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3909 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)(8,13,15,27)(10,21)(14,26,29,31)(16,33,32,28)(17,30,34,38)(19,35)(22,36)(23,37);(7,8,10,16,11,13,23,28,12,15,21,32,25,27,37,33)(9,17,19,14,20,30,22,31,18,34,35,29,24,38,36,26);(7,9)(8,14,15,29)(10,22)(11,24)(12,18)(13,26,27,31)(16,34,32,17)(19,23)(20,25)(21,36)(28,30,33,38)(35,37);(4,6)(8,15)(9,18)(13,27)(16,32)(19,35)(20,24)(22,36)(28,33)

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

C2×#498377 #72320624