C₂×#495897 #72320646

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

Aliases

direct product

C₂×#495897

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 ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4035 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)(8,13,15,26)(10,21)(11,24)(14,25,29,30)(16,33,32,27)(17,34,31,38)(19,28)(20,35)(22,36)(23,37);(7,8,10,13,11,16,21,27,12,15,23,26,24,32,37,33)(9,17,19,34,20,14,28,30,18,31,22,38,35,29,36,25);(7,9)(8,14,15,29)(10,22)(11,20)(12,18)(13,25,26,30)(16,31,32,17)(19,23)(21,36)(24,35)(27,34,33,38)(28,37);(4,6)(8,15)(9,18)(13,26)(16,32)(19,22)(20,35)(27,33)(28,36)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72250872, C₂×#72250867
maximal normal subgroup of	C₂×#72250872, C₂×#72250867

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

C2×#495897 #72320646