#193157

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#385
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100353);