#437796

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1256 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)(6,8)(9,10)(11,23)(12,29)(13,31)(14,20)(15,22)(16,21)(17,26)(18,30)(19,33)(24,35)(25,39)(27,36)(28,37)(32,34)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,27,14,26,16,28)(10,17,22,37,20,23,21,36)(12,25,30,40,13,32,33,24)(18,35,31,39,19,38,29,34);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,18,20,19)(11,24,26,40)(15,33,16,30)(17,34,23,39)(21,31,22,29)(25,27,32,28)(35,37,38,36);(2,5)(6,8)(9,13,14,12)(10,19,20,18)(11,25,26,32)(15,33,16,30)(17,35,23,38)(21,31,22,29)(24,27,40,28)(34,37,39,36)

Polycyclic presentation:

2^9|b^2=g,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,g^c=gi,h^d=hi

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, 344992);