C₂×#8799 #191258

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×#8799

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

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, 98454);