#332419

order	512
nilpotency class	3
isoclinism	isoclinic class of #332327
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

579 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)(4,7,14,8)(6,13,15,11)(17,18)(19,28)(20,24)(21,25)(22,27)(23,31)(26,36)(29,44)(30,38)(32,42)(33,43)(34,39)(35,41)(37,45)(40,47)(46,48);(17,19,22,32)(18,23,27,38)(20,33,35,29)(21,30,36,31)(24,37,41,48)(25,28,26,42)(34,46,47,45)(39,43,40,44);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,20,22,35)(18,24,27,41)(19,29,32,33)(21,34,36,47)(23,37,38,48)(25,39,26,40)(28,43,42,44)(30,45,31,46);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(18,25)(19,30)(21,36)(24,39)(26,27)(28,42)(29,45)(31,32)(33,46)(34,47)(40,41)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,e^2=h,f^2=i,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,f^c=fi,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, 239615);