#169798

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,8)(3,7,4,6)(10,13)(11,14)(15,16)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(2,6)(3,4)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(9,11)(10,14)(12,15)(13,16)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31);(1,3,5,4)(2,6,8,7)(18,24)(22,28)(23,32)(30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gh,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, 76994);