#199220

order	512
nilpotency class	3
isoclinism	isoclinic class of #199212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106416);