C₂×#7471 #151764

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7471

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	D₁₆
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 258 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 — 3 generators on 42 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58960);