#175928

order	512
nilpotency class	3
isoclinism	isoclinic class of #175533
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

824 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,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, 83124);