C₂×#13723 #513033

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#13723

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2246	C₂
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

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

normal subgroups only

This group has 442 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) = 42
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420229);