#478282

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1005 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385478);