#186313

order	512
nilpotency class	3
isoclinism	isoclinic class of #186146
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

2181 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239884
maximal normal subgroup of	#72239884
maximal quotient of	#72239899

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93509);