#203125

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

687 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh,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, 110321);