#332046

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3214	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

507 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)(20,33)(21,23)(22,27)(24,31)(25,35)(26,36)(29,43)(30,41)(32,39)(34,38)(37,42)(40,44)(45,47)(46,48);(1,3)(2,5)(4,9)(6,8)(7,10)(11,13)(12,15)(14,16)(18,23)(20,34)(21,36)(24,39)(25,33)(26,27)(28,41)(29,44)(30,31)(35,37)(43,48)(45,46);(2,8)(3,10)(6,7)(9,16)(11,12)(14,15)(17,19)(18,24)(20,29)(21,30)(22,32)(23,28)(25,40)(26,39)(27,41)(31,36)(33,47)(34,44)(35,46)(37,45)(38,43)(42,48);(1,4)(2,9)(3,11)(5,13)(6,15)(7,14)(8,16)(10,12)(17,20,22,35)(18,25,27,33)(19,29,32,46)(21,37,36,34)(23,38,26,42)(24,40,41,47)(28,43,39,48)(30,45,31,44)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239242);