#429600

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

962 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 336796);