#430337

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=efi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 337533);