#186126

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62781	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2271 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93322);