C₂×#494893 #72320767

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#494893

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5165 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);(7,8)(9,15)(10,25)(11,17)(12,18)(13,19)(14,20)(16,24)(21,29)(22,30)(23,31)(26,34)(27,35)(28,36)(32,38)(33,37);(3,4)(5,6)(7,9)(8,15)(10,12)(11,21)(13,24)(14,22)(16,19)(17,29)(18,25)(20,30)(23,27)(26,38)(28,37)(31,35)(32,34)(33,36);(9,21,22,37)(10,23,24,38)(11,26)(12,13)(15,29,30,33)(16,32,25,31)(17,34)(18,19)(27,28)(35,36);(4,6)(7,10,12,22,14,24,13,9)(8,16,18,15,20,25,19,30)(11,23,26,21,28,38,27,37)(17,32,34,33,36,31,35,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72239538, #72343773, #72343897
maximal normal subgroup of	C₂×#72239538, #72343773, #72343897

GAP

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

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

C2×#494893 #72320767