#6452257

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3255 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,16)(13,26)(14,17)(15,22)(18,28)(20,34)(21,35)(23,39)(24,32)(25,33)(27,31)(29,38)(30,37)(36,40);(9,11,13,22,14,21,28,37)(10,15,18,19,17,30,26,35)(12,23,25,38,24,20,27,36)(16,29,31,39,32,40,33,34);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,20)(13,27)(14,24)(15,29)(17,32)(18,33)(19,34)(21,23)(22,38)(25,28)(26,31)(30,40)(35,39)(36,37);(1,4,5,3)(2,6,7,8)(11,21)(13,28)(18,26)(19,35)(20,23)(25,27)(31,33)(34,39);(10,17)(12,24)(15,30)(18,26)(19,35)(20,23)(25,27)(36,38)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,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,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, 6359453);