#6440847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
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	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2593 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)(9,10,14,20)(11,17,24,32)(12,25,29,37)(13,18,30,19)(15,23,33,22)(16,27,36,28)(21,38,40,31)(26,35,39,34);(2,5)(6,8)(9,11,13,22,14,24,30,23)(10,15,19,17,20,33,18,32)(12,26,28,40,29,39,27,21)(16,34,25,31,36,35,37,38);(1,3,4,7)(2,6,5,8)(9,12,13,28,14,29,30,27)(10,16,19,25,20,36,18,37)(11,21,22,26,24,40,23,39)(15,31,17,35,33,38,32,34);(9,11)(10,17)(12,21)(13,22)(14,24)(15,18)(16,35)(19,33)(20,32)(23,30)(25,38)(26,28)(27,39)(29,40)(31,37)(34,36);(9,13,14,30)(10,18,20,19)(11,22,24,23)(12,27,29,28)(15,32,33,17)(16,25,36,37)(21,39,40,26)(31,35,38,34)

Polycyclic presentation:

2^9|a^2=i,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,h^b=hi,e^c=ei,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, 6348043);