#332042

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
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	#19509	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

524 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,e^2=h,f^2=i,g^2=hi,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=ghi,f^c=fi,g^c=ghi,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, 239238);