#199229

order	512
nilpotency class	3
isoclinism	isoclinic class of #199221
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

1067 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)(7,11)(12,16)(13,15)(17,18)(19,23,28,38)(20,30)(21,34)(22,25)(24,32)(26,36)(27,43,31,48)(29,46,45,39)(33,42)(35,41)(37,47,40,44);(1,3,5,11,6,13,4,12)(2,7,8,10,9,16,14,15)(17,19)(18,23,25,38)(20,31)(21,29)(22,28)(24,40,41,37)(26,39,34,46)(27,33)(30,48,42,43)(32,47)(35,44)(36,45);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,22,33)(18,24,25,41)(19,27,28,31)(21,35,36,32)(23,37,38,40)(26,42,34,30)(29,47,45,44)(39,48,46,43);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(18,25)(23,38)(24,41)(26,34)(30,42)(37,40)(39,46)(43,48)

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,c^b=ch,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106425);