#6453866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437452
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62851	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3085 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)(11,20)(12,18)(13,19)(14,27)(15,17)(16,22)(21,36)(23,31)(24,39)(25,34)(26,33)(28,32)(29,35)(30,37)(38,40);(9,11,15,23)(10,16,17,32)(12,24,26,21)(13,22,14,28)(18,30,33,40)(19,31,27,20)(25,38,29,37)(34,39,35,36);(2,5)(6,8)(10,17)(13,14)(18,33)(20,31)(22,28)(25,29)(36,39)(37,38);(9,12,15,26)(10,18,17,33)(11,21,23,24)(13,25,14,29)(16,30,32,40)(19,34,27,35)(20,36,31,39)(22,37,28,38);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,19,17,27)(11,22,23,28)(12,25,26,29)(16,31,32,20)(18,34,33,35)(21,37,24,38)(30,39,40,36)

Polycyclic presentation:

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

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, 6361062);