C₂×#7420 #150246

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

Aliases

direct product

C₂×#7420

Subgroups

	subgroup	G/H
center, socle	C₂³	#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 90 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)(5,9,10,7)(6,8)(11,12,14,21,18,24,17,35)(13,25,28,41,31,19,30,36)(15,33,27,23,16,34,32,22)(20,39,37,26,40,42,38,29);(3,5)(4,7)(6,10)(8,9)(12,19)(13,26)(14,17)(15,32)(16,27)(20,23)(21,41)(22,40)(24,25)(28,39)(29,31)(30,42)(33,37)(34,38)(35,36)

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, 57442);