#434713

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 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)(9,10,15,20,16,11,14,19)(12,17,26,33,27,21,25,23)(13,28,31,22,30,34,29,18)(24,37,40,32,39,36,38,35);(1,2)(3,6)(4,5)(7,8)(9,11,16,10)(12,23,27,33)(13,28,30,34)(14,19,15,20)(17,26,21,25)(18,29,22,31)(24,32,39,35)(36,40,37,38);(1,3,4,7)(2,6,5,8)(9,12,14,25,16,27,15,26)(10,17,19,23,11,21,20,33)(13,24,31,40,30,39,29,38)(18,32,28,36,22,35,34,37);(2,5)(6,8)(9,13)(10,18)(11,22)(12,24)(14,31)(15,29)(16,30)(17,32)(19,28)(20,34)(21,35)(23,36)(25,40)(26,38)(27,39)(33,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,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, 341909);