#113185

order	512
nilpotency class	3
isoclinism	isoclinic class of #113183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20381);