#108479

order	512
nilpotency class	3
isoclinism	isoclinic class of #106004
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,e^a=eh,g^a=gh,f^b=fh,e^c=ei,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15675);