C₂×#8797 #191253

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#8797

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 520 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) = 50
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98449);