#178728

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46016	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(4,11)(6,13)(7,12)(9,15)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,2,5,8)(3,6,10,13)(4,12,11,7)(9,16,15,14)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31)(33,35,37,43)(34,38,40,46)(36,42,44,48)(39,45,41,47);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,15)(12,16)(20,21)(24,25)(27,32)(28,30)(34,39)(36,44)(38,45)(40,41)(42,48)(46,47);(1,4,5,11)(2,7,8,12)(3,9,10,15)(6,14,13,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)(34,40)(38,46)(39,41)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85924);