C₂×#13720 #513029

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#13720

Subgroups

	subgroup	G/H
center, socle, prime radical	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

410 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 42
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,19,13)(14,30,32,36)(15,27,35,23)(16,26,20,22)(17,37,38,42)(18,29,21,25)(24,40,28,33)(31,34,41,39);(4,7)(8,10)(11,13,16,27,17,28,34,32,21,25,30,41,40,37,15,26,19,12,20,23,38,24,39,14,18,29,36,31,33,42,35,22);(3,5,6,9)(4,8,7,10)(11,14,15,28,33,23,30,13,18,26,34,42,38,41,16,29,19,32,35,24,40,27,36,12,21,22,39,37,17,31,20,25)

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, 420225);