#489397

order	512
nilpotency class	4
isoclinism	isoclinic class of #489394
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	#62428	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 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)(9,10,14,18)(11,19,21,37)(12,23,25,39)(13,28,26,17)(15,30,31,20)(16,33,35,27)(22,38,29,34)(24,36,40,32);(3,7)(5,8)(9,11)(10,15)(12,24)(13,29)(14,21)(16,34)(17,36)(18,31)(19,23)(20,27)(22,26)(25,40)(28,32)(30,33)(35,38)(37,39);(1,2)(3,8)(4,6)(5,7)(12,25)(16,35)(19,37)(20,30)(23,39)(24,40)(27,33)(34,38);(9,12,26,33,14,25,13,27)(10,16,28,39,18,35,17,23)(11,20,29,40,21,30,22,24)(15,19,36,38,31,37,32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396593);