#331948

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

501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,10)(3,11,14,27)(4,15,18,28)(5,19,22,9)(7,20,23,13)(8,21,26,17)(12,30,31,24)(16,25,32,29)(33,34,39,44)(35,45,48,53)(36,42,51,57)(37,40,38,43)(41,52,56,46)(47,61,63,59)(49,58,50,54)(55,60,64,62);(1,3)(2,7)(4,16)(5,20)(6,14)(8,25)(9,27)(10,23)(11,19)(12,17)(13,22)(15,24)(18,32)(21,31)(26,29)(28,30)(34,40)(35,46)(36,49)(37,38)(42,57)(43,44)(45,53)(48,52)(50,51)(55,61)(59,64)(60,62);(7,23)(11,27)(24,30)(25,29)(33,35,39,48)(34,41,44,56)(36,47,51,63)(37,46,38,52)(40,53,43,45)(42,55,57,64)(49,62,50,60)(54,61,58,59);(1,4,6,18)(2,8,10,26)(3,12,14,31)(5,21,22,17)(7,24,23,30)(9,28,19,15)(11,29,27,25)(13,32,20,16)(33,36)(34,42)(35,47)(37,50)(38,49)(39,51)(40,54)(41,55)(43,58)(44,57)(45,59)(46,60)(48,63)(52,62)(53,61)(56,64)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,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, 239144);