#179120

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,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, 86316);