#331951

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	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

507 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:

(1,2,6,11)(3,12,16,26)(4,8,19,25)(5,9,20,10)(7,14,22,15)(13,28,17,27)(18,23,31,24)(21,30,32,29)(33,34,39,44)(35,45,49,56)(36,50,52,42)(37,40,38,43)(41,46,57,48)(47,59,62,64)(51,54,53,58)(55,60,63,61);(1,3)(2,7)(4,17)(5,14)(6,16)(8,21)(9,12)(10,26)(11,22)(13,19)(15,20)(18,30)(23,27)(24,28)(25,32)(29,31)(34,40)(35,46)(36,51)(37,38)(42,50)(43,44)(45,56)(48,49)(52,53)(55,59)(60,61)(63,64);(1,4,6,19)(2,8,11,25)(3,13,16,17)(5,18,20,31)(7,21,22,32)(9,23,10,24)(12,27,26,28)(14,29,15,30)(33,35)(34,41)(36,47)(37,46)(38,48)(39,49)(40,45)(42,55)(43,56)(44,57)(50,63)(51,61)(52,62)(53,60)(54,59)(58,64);(2,9)(3,14)(5,20)(8,23)(10,11)(12,26)(13,29)(15,16)(17,30)(18,31)(24,25)(27,28)(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|a^2=h,c^2=hi,d^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, 239147);