#334720

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

605 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,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, 241916);