#6448776

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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3156 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)(11,22)(12,18)(13,19)(14,20)(15,29)(16,21)(17,25)(23,38)(24,32)(26,33)(27,35)(28,37)(30,36)(31,40)(34,39);(9,11,16,12)(10,17,21,32)(13,23,28,26)(14,25,15,24)(18,29,22,20)(19,34,37,31)(27,40,30,39)(33,35,38,36);(2,5)(6,8)(9,12,16,11)(10,18,21,22)(13,26,28,23)(14,24,15,25)(17,20,32,29)(19,33,37,38)(27,39,30,40)(31,35,34,36);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,23)(12,26)(14,27)(15,30)(16,28)(17,31)(18,33)(20,35)(21,37)(22,38)(24,39)(25,40)(29,36)(32,34);(9,14,16,15)(10,20,21,29)(11,24,12,25)(13,27,28,30)(17,22,32,18)(19,35,37,36)(23,39,26,40)(31,38,34,33)

Polycyclic presentation:

2^9|b^2=i,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,e^b=ei,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, 6355972);