C₂×#13653 #512759

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13653

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 70 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419955);