#199511

order	512
nilpotency class	3
isoclinism	isoclinic class of #199251
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1046 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,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, 106707);