C₂×#7446 #150942

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7446

Subgroups

	subgroup	G/H
center, socle	C₂³	#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

296 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256

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

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58138);