#210392

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#18984	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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)(3,7)(6,8)(9,10,14,21,16,22,15,20)(11,23,27,37,28,30,26,17)(12,29,31,18,32,25,36,38)(13,19,33,40,35,24,34,39);(3,7)(5,8)(9,11)(10,17)(12,24)(13,25)(14,27)(15,26)(16,28)(18,34)(19,32)(20,30)(21,23)(22,37)(29,35)(31,39)(33,38)(36,40);(1,2)(3,8)(4,6)(5,7)(9,12)(10,18)(11,24)(13,23)(14,31)(15,36)(16,32)(17,34)(19,28)(20,29)(21,25)(22,38)(26,40)(27,39)(30,35)(33,37);(9,13,14,33,16,35,15,34)(10,19,21,40,22,24,20,39)(11,25,27,38,28,29,26,18)(12,30,31,17,32,23,36,37)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117588);