C₂×#7405 #150187

order	512
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7405

Subgroups

	subgroup	G/H
center, socle	C₂³	#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

650 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,14,19,16,21,28,38)(13,22,23,36,25,33,40,41)(15,20,30,37,31,18,27,35)(17,26,32,29,34,39,42,24);(4,7)(5,10)(8,9)(11,13,16,25)(12,17,21,34)(14,26,28,39)(15,29,31,24)(18,33,20,22)(19,36,38,41)(23,27,40,30)(32,37,42,35)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57383);