#108047

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

839 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15243);