#178318

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
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

654 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85514);