C₂×#493158 #72343533

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#493158

Subgroups

	subgroup	G/H
center	C₂²×C₄	#442
commutator	C₈	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂³	C₂×#442
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4497 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72239482, #72343873
maximal normal subgroup of	C₂×#72239482, #72343873

GAP

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

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

C2×#493158 #72343533