C₂×#10517 #238807

order	512
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10517

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 128

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146003);