C₂×#7386 #150061

order	512
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7386

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#357
commutator	C₁₆	C₂²×C₈
Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₁₆ ▷ 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

262 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 512

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#150063, C₂×#151083
maximal normal subgroup of	C₂×#150063, C₂×#151083

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57257);