C₂×#10217 #232858

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10217

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 202 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)(5,12)(6,8)(7,17)(9,11)(10,13)(14,18)(15,16)(19,20,25,30)(21,31,34,45)(22,35,38,43)(23,29,24,28)(26,39,41,33)(27,36,44,37)(32,40,48,42)(46,49,50,47);(3,5,8,14)(4,10,6,16)(7,15,9,13)(11,18,17,12)(19,21,25,34)(20,26,30,41)(22,32,38,48)(23,39,24,33)(27,42,44,40)(28,45,29,31)(35,46,43,50)(36,49,37,47);(3,6,8,4)(5,12,14,18)(7,17,9,11)(10,13,16,15)(19,22,25,38)(20,27,30,44)(21,32,34,48)(23,37,24,36)(26,40,41,42)(28,35,29,43)(31,46,45,50)(33,49,39,47)

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, 140054);