#199959

order	512
nilpotency class	3
isoclinism	isoclinic class of #199806
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1001 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107155);