#117859

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

629 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,d^c=di,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25055);