#197194

order	512
nilpotency class	3
isoclinism	isoclinic class of #197026
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

1027 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:

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

Polycyclic presentation:

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

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, 104390);