#108171

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,11,7,10,6,12)(3,4,14,15,13,8,9,16)(17,18,22,26,23,28,21,27)(19,29,31,24,32,39,37,40)(20,33,35,25,36,41,34,42)(30,44,45,48,46,38,47,43);(1,3,6,14,7,13,5,9)(2,8,12,16,10,4,11,15)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,38)(26,39)(27,29)(28,40)(33,48)(34,47)(35,45)(36,46)(41,43)(42,44);(1,4,7,8)(2,9,10,14)(3,11,13,12)(5,16,6,15)(17,20,23,36)(18,25,28,42)(19,30,32,46)(21,35,22,34)(24,38,40,44)(26,33,27,41)(29,43,39,48)(31,47,37,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15367);