#331945

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	#18854	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,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, 239141);