#488976

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61088	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

907 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)(9,10,13,16)(11,19,22,37)(12,23,27,39)(14,29,26,18)(15,31,32,20)(17,34,25,28)(21,38,30,35)(24,36,40,33);(2,6)(3,4)(7,8)(9,11)(10,15)(12,24)(13,22)(14,30)(16,32)(17,35)(18,36)(19,23)(20,28)(21,26)(25,38)(27,40)(29,33)(31,34)(37,39);(2,7)(3,4)(6,8)(9,10)(11,15)(12,25)(13,16)(14,18)(17,27)(19,31)(20,37)(21,33)(22,32)(23,34)(24,38)(26,29)(28,39)(30,36)(35,40);(10,16)(15,32)(17,25)(18,29)(19,37)(23,39)(33,36)(35,38)

Polycyclic presentation:

2^9|a^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,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, 396172);