#172126

order	512
nilpotency class	3
isoclinism	isoclinic class of #171725
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

739 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgh,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, 79322);