#193140

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
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

814 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:

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

Polycyclic presentation:

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

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, 100336);