#332836

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

573 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)(3,7,5,9)(4,12,13,15)(8,14,16,11)(17,18)(19,29)(20,34)(21,24)(22,27)(23,28)(25,32)(26,37)(30,41)(31,43)(33,40)(35,42)(36,38)(39,46)(44,47)(45,48);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(18,24)(19,30)(20,35)(21,22)(27,28)(29,40)(31,45)(32,33)(36,37)(38,42)(39,48)(43,47);(1,4,6,13)(2,8,10,16)(3,11,5,14)(7,15,9,12)(17,19,23,33)(18,25,28,41)(20,31,37,45)(21,30,22,32)(24,29,27,40)(26,39,34,47)(35,44,36,46)(38,43,42,48);(2,9)(3,5)(7,10)(8,12)(11,14)(15,16)(17,20,23,37)(18,26,28,34)(19,31,33,45)(21,35,22,36)(24,38,27,42)(25,39,41,47)(29,43,40,48)(30,44,32,46)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240032);