#172139

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29622	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1167 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,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,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79335);