C₂×#7538 #153554

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7538

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₃₂
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
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

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

normal subgroups only

This group has 186 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) = 512

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

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

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, 60750);