#169545

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=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, 76741);