#331967

order	512
nilpotency class	3
isoclinism	isoclinic class of #331927
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	#18857	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

528 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:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239163);