#170116

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
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

683 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18,23,20)(19,26,30,32)(21,25,22,31)(24,28,27,29)(33,34,38,41)(35,39,37,45)(36,43,44,47)(40,46,48,42);(1,3,5,10)(2,6,7,15)(4,13,14,11)(8,16,12,9)(17,19,23,30)(18,24,20,27)(21,29,22,28)(25,32,31,26)(33,35,38,37)(34,39,41,45)(36,42,44,46)(40,47,48,43);(17,20,23,18)(19,27,30,24)(21,31,22,25)(26,28,32,29)(33,36,38,44)(34,40,41,48)(35,42,37,46)(39,47,45,43);(2,7)(3,10)(8,12)(11,13)(17,21,23,22)(18,25,20,31)(19,28,30,29)(24,26,27,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77312);