#186338

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2026 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:

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

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=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, 93534);