#6440842

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60369	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348038);