#203121

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110317);