#193406

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

793 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fi

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, 100602);