#339878

order	512
nilpotency class	3
isoclinism	isoclinic class of #339847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

511 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247074);