C₂×#7406 #150192

order	512
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7406

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂²	#1343
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

629 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 74 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);(3,4,7,12)(5,15,17,44)(6,11,22,35)(8,28,27,58)(9,13,26,41)(10,30,31,20)(14,42,40,21)(16,33,46,38)(18,51,50,66)(19,45,49,64)(23,37,55,39)(24,56,54,65)(25,57,29,34)(32,59,60,52)(36,62,61,53)(43,48,63,47);(3,5)(4,10)(6,20)(7,25)(8,29)(9,19)(11,34)(12,38)(13,32)(14,43)(15,36)(16,24)(17,42)(18,21)(22,48)(23,52)(26,50)(27,49)(28,33)(30,54)(31,40)(35,44)(37,51)(39,64)(41,63)(45,56)(46,55)(47,53)(57,65)(58,60)(59,62)(61,66)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57388);