C₂×#7537 #153553

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7537

Subgroups

	subgroup	G/H
center, socle	C₂³	D₃₂
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	C₂²×D₃₂	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

206 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 70
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 70 points:

(1,2);(3,4,5,6)(7,8)(9,14)(10,25)(11,29)(12,32)(13,19)(15,20)(16,21)(17,26)(18,30)(22,42)(23,44)(24,36)(27,60)(28,58)(31,62)(33,46)(34,45)(35,53)(37,52)(38,55)(39,48)(40,61)(41,51)(43,59)(47,63)(49,57)(50,56)(54,67)(64,70)(65,69)(66,68);(8,14)(9,20)(10,21)(11,26)(12,30)(15,16)(17,42)(18,44)(19,36)(22,52)(23,55)(24,48)(25,33)(27,61)(28,51)(29,34)(31,59)(32,35)(37,57)(38,56)(39,41)(40,63)(43,67)(45,46)(47,53)(49,50)(54,70)(58,65)(60,64)(62,66)(68,69)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60749);