#332690

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

558 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=hi,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=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239886);