C₂×#493228 #72320514

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#493228

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂³	C₂²×#97
prime radical	C₂³×D₄	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4337 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);(7,8,10,15,11,16,22,32,12,14,23,29,24,30,37,38)(9,17,19,33,20,13,25,27,18,28,21,35,34,26,36,31);(7,9)(8,13,14,26)(10,21)(11,20)(12,18)(15,31,29,27)(16,28,30,17)(19,23)(22,36)(24,34)(25,37)(32,33,38,35);(4,6)(8,14)(9,18)(15,29)(16,30)(19,21)(20,34)(25,36)(32,38)

Conjugacy classes & character table

220 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), (7,8,10,15,11,16,22,32,12,14,23,29,24,30,37,38)(9,17,19,33,20,13,25,27,18,28,21,35,34,26,36,31), (7,9)(8,13,14,26)(10,21)(11,20)(12,18)(15,31,29,27)(16,28,30,17)(19,23)(22,36)(24,34)(25,37)(32,33,38,35), (4,6)(8,14)(9,18)(15,29)(16,30)(19,21)(20,34)(25,36)(32,38));

C2×#493228 #72320514