#331934

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	#3245	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

514 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)(3,11)(4,15)(5,19)(6,10)(7,13)(8,17)(9,22)(12,30)(14,27)(16,25)(18,28)(20,23)(21,26)(24,31)(29,32)(33,34)(35,45)(36,50)(37,40)(38,43)(39,44)(41,48)(42,52)(46,57)(47,59)(49,56)(51,54)(53,58)(55,61)(60,63)(62,64);(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)(36,51)(37,38)(42,50)(43,44)(45,56)(48,49)(52,53)(55,59)(60,61)(63,64);(7,23)(11,27)(24,30)(25,29)(33,35,39,49)(34,41,44,57)(36,47,52,62)(37,46,38,48)(40,45,43,56)(42,55,50,63)(51,61,53,60)(54,59,58,64);(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)(33,36,39,52)(34,42,44,50)(35,47,49,62)(37,53,38,51)(40,54,43,58)(41,55,57,63)(45,59,56,64)(46,60,48,61)

Polycyclic presentation:

2^9|c^2=i,d^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, 239130);