#169352

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

686 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)(3,9,10,16)(4,12,14,7)(6,13,15,11)(18,23)(20,21)(24,31)(25,26)(27,29)(30,32)(33,34,38,41)(35,39,37,45)(36,43,44,47)(40,46,48,42);(1,3,5,10)(2,6,8,15)(4,13,14,11)(7,16,12,9)(17,18,22,26)(19,24,28,32)(20,23,21,25)(27,30,29,31)(33,35,38,37)(34,39,41,45)(36,42,44,46)(40,47,48,43);(17,19,22,28)(18,24,26,32)(20,27,21,29)(23,30,25,31)(33,36,38,44)(34,40,41,48)(35,42,37,46)(39,47,45,43);(3,10)(6,15)(9,16)(11,13)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=h,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, 76548);