#334819

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

628 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=hi,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, 242015);