#6809301

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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	#61057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2147 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)(11,17)(12,15)(13,18)(14,26)(16,19)(20,23)(21,32)(22,33)(24,31)(25,29)(27,34)(28,39)(30,35)(36,37)(38,40);(2,5)(3,7)(9,11,16,23)(10,17,19,20)(12,24,26,33)(13,27,30,40)(14,22,15,31)(18,34,35,38)(21,28,37,29)(25,32,39,36);(9,12,14,10,16,26,15,19)(11,20,31,33,23,17,22,24)(13,25,28,18,30,39,29,35)(21,36,40,34,37,32,27,38);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,13,14,28,16,30,15,29)(10,18,26,39,19,35,12,25)(11,21,31,40,23,37,22,27)(17,32,24,38,20,36,33,34)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi

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, 6716497);