#113194

order	512
nilpotency class	3
isoclinism	isoclinic class of #113192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh,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, 20390);