#107524

order	512
nilpotency class	3
isoclinism	isoclinic class of #8510
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

893 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)(11,20,23,16)(12,24)(13,27)(14,19)(15,18)(17,26)(21,36,37,40)(22,31,28,32)(25,33)(29,34)(30,38,35,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,22,15,23,13,28)(10,16,19,32,18,20,27,31)(12,21,26,39,25,37,29,38)(17,30,33,40,34,35,24,36);(3,8)(5,7)(9,12)(10,17)(11,21)(13,29)(14,26)(15,25)(16,30)(18,34)(19,33)(20,35)(22,39)(23,37)(24,27)(28,38)(31,36)(32,40)

Polycyclic presentation:

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

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, 14720);