#332022

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:

(1,2,8,14)(3,15,19,29)(4,10,5,23)(6,13,7,12)(9,20,28,18)(11,22,31,25)(16,27,17,30)(21,32,24,26)(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)(2,9)(4,21)(5,24)(6,20)(7,18)(8,19)(10,30)(11,26)(12,29)(13,15)(14,28)(16,22)(17,25)(23,27)(31,32)(34,40)(36,49)(37,38)(41,45)(42,57)(43,44)(46,61)(47,52)(50,51)(53,56)(59,60)(62,63);(1,4,8,5)(2,10,14,23)(3,16,19,17)(6,22,7,25)(9,26,28,32)(11,13,31,12)(15,30,29,27)(18,21,20,24)(33,35)(34,41)(36,46)(37,47)(38,52)(39,48)(40,53)(42,55)(43,45)(44,56)(49,61)(50,63)(51,62)(54,60)(57,64)(58,59);(1,5,8,4)(2,11,14,31)(3,17,19,16)(6,22,7,25)(9,27,28,30)(10,13,23,12)(15,32,29,26)(18,21,20,24)(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=hi,d^2=h,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, 239218);