#179593

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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	#29622	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1095 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)(10,14)(11,12)(15,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(1,3)(2,5)(4,7)(6,8)(11,12)(14,15)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(9,11,13,12)(10,15,16,14)(18,23)(22,30)(24,31)(29,32)

Polycyclic presentation:

2^9|c^2=i,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86789);