C₂×#8798 #191257

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#8798

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

500 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 242 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);(3,4,8,12,9,13,7,17)(5,14,15,18,16,11,6,10)(19,20,24,30,25,28,23,29)(21,31,33,45,34,26,39,42)(22,35,37,44,38,46,40,27)(32,47,49,43,36,50,48,41);(3,5,7,6,9,16,8,15)(4,10,12,14,13,18,17,11)(19,21,23,33,25,34,24,39)(20,26,29,42,28,31,30,45)(22,36,37,49,38,32,40,48)(27,43,35,47,44,41,46,50);(3,6,7,16,9,15,8,5)(4,11,12,10,13,14,17,18)(19,22)(20,27,28,44)(21,32)(23,37)(24,40)(25,38)(26,41,31,43)(29,35,30,46)(33,48)(34,36)(39,49)(42,50,45,47)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98453);