#6439376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23409	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2063 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,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, 6346572);