C₂×#7384 #150055

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7384

Subgroups

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

266 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 114 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)(13,21)(14,23)(15,27)(18,33)(19,36)(20,32)(22,26)(24,40)(25,30)(28,34)(29,31)(35,41)(37,38)(39,42);(11,12,16,20)(13,22,25,39)(14,26,28,42)(15,29,27,19)(17,30,32,21)(18,34,35,23)(24,41,37,33)(31,38,36,40)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#150057, C₂×C₈×SD₆₄, C₂×#151078
maximal normal subgroup of	C₂×#150057, C₂×C₈×SD₆₄, C₂×#151078

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57251);