#436909

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

991 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)(11,17)(12,26)(13,31)(14,20)(15,22)(16,21)(18,28)(19,25)(23,38)(24,33)(27,35)(29,37)(30,36)(32,40)(34,39);(1,3,4,8)(2,5,6,7)(9,11,15,13,14,25,16,24)(10,17,21,33,20,19,22,31)(12,27,28,23,30,40,29,39)(18,34,26,32,37,38,36,35);(2,6)(5,7)(9,12)(10,18)(11,23)(13,27)(14,30)(15,29)(16,28)(17,32)(19,35)(20,37)(21,36)(22,26)(24,40)(25,39)(31,38)(33,34);(3,8)(5,7)(9,13,16,25,14,24,15,11)(10,19,22,31,20,17,21,33)(12,27,28,39,30,40,29,23)(18,35,26,38,37,32,36,34)

Polycyclic presentation:

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

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, 344105);