#186530

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239103, #72239880
maximal normal subgroup of	#72239103, #72239880
maximal quotient of	#72239143, #72239145, #72239900

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93726);