#332008

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	#19509	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

519 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=hi,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, 239204);