#6450174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
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	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3002 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:

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

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,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6357370);