#186343

order	512
nilpotency class	3
isoclinism	isoclinic class of #186162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62836	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2233 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93539);