#6450178

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2917 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6357374);