#186291

order	512
nilpotency class	3
isoclinism	isoclinic class of #186137
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2273 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi,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, 93487);