#332019

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	#3207	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,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, 239215);