#186130

order	512
nilpotency class	3
isoclinism	isoclinic class of #186128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3518	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,f^c=fh,g^c=gh

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, 93326);