C₂×#13685 #512907

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13685

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×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

546 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(4,6)(7,8)(9,13)(10,22)(11,25)(12,17)(14,18)(15,19)(16,23)(20,32)(21,28)(24,38)(26,34)(27,33)(29,37)(30,35)(31,36);(3,4,5,6);(8,13)(9,18)(10,19)(11,23)(14,15)(16,32)(17,28)(20,37)(21,35)(22,26)(24,36)(25,27)(29,38)(30,31)(33,34)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#513003, #72322876, #72322886
maximal normal subgroup of	C₂×#513003, #72322876, #72322886
maximal quotient of	#72321817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420103);