#507971

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi,f^d=fi

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, 415167);