#6499847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499844
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62821	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3872 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258210
maximal normal subgroup of	#72258210
maximal quotient of	#72258260, #72258264

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407043);