#109075

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

608 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:

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

Polycyclic presentation:

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

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, 16271);