#333180

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

571 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240376);