#488967

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	#61038	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

926 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 396163);