#108673

order	512
nilpotency class	3
isoclinism	isoclinic class of #108672
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,f^b=fi,e^c=ei,f^c=fh

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, 15869);