C₂×#10529 #238931

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10529

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 830 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) = 50
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(3,4)(5,13)(6,11)(7,10)(8,12)(9,15)(14,18)(16,17)(19,20,25,29)(21,30,34,45)(22,35,38,27)(23,40,24,28)(26,33,42,32)(31,46,48,43)(36,41,47,50)(37,44,39,49);(3,5)(4,9)(6,15)(7,16)(8,14)(10,18)(11,13)(12,17)(19,21,25,34)(20,26,29,42)(22,36,38,47)(23,33,24,32)(27,43,35,46)(28,45,40,30)(31,37,48,39)(41,49,50,44);(4,10)(5,14)(7,8)(11,12)(15,16)(17,18)(19,22,23,37,25,38,24,39)(20,27,28,44,29,35,40,49)(21,31,32,47,34,48,33,36)(26,41,30,43,42,50,45,46)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146127);