#394963

order	512
nilpotency class	3
isoclinism	isoclinic class of #394098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1518 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,25)(14,18)(17,22)(19,28)(20,32)(21,30)(23,36)(24,34)(26,35)(27,31)(29,37)(33,38)(39,40);(2,5)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,32)(17,25)(18,30)(19,31)(23,38)(24,37)(26,39)(27,28)(29,34)(33,36)(35,40);(9,12,14,24)(10,16,18,34)(11,20,21,37)(13,26,28,23)(15,29,30,32)(17,33,31,40)(19,36,25,35)(22,39,27,38);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,20)(13,27)(14,21)(15,25)(16,33)(18,31)(19,30)(22,28)(23,39)(24,37)(26,38)(29,35)(32,36)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302159);