C₂×#7382 #150052

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7382

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	D₄⋊C₁₆
prime radical	C₂×#135	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 250 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

(1,2);(3,4,7,14,9,13,30,41,10,15,31,44,6,23,25,53,11,16,32,45,26,42,56,65,8,33,29,60,27,46,57,66)(5,17,19,47,21,12,35,36,20,38,52,62,24,39,54,63,22,49,28,58,18,40,50,43,34,48,59,64,51,37,55,61);(3,5)(4,12)(6,24)(7,28)(8,34)(9,21)(10,20)(11,22)(13,38)(14,43)(15,39)(16,40)(17,46)(18,26)(19,32)(23,49)(25,55)(27,51)(29,52)(30,50)(31,59)(33,37)(35,56)(36,45)(41,64)(42,48)(44,61)(47,53)(54,57)(58,66)(60,63)(62,65)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#151103, D₃×#7382, #72287101, …
maximal normal subgroup of	C₂×#151103, #72287101, #72287225, …
maximal quotient of	D₃×#7382, #72287229

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57248);