#108325

order	512
nilpotency class	3
isoclinism	isoclinic class of #108316
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

936 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:

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

Polycyclic presentation:

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

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, 15521);