#6448775

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	#60423	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3218 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)(11,17)(12,18)(13,19)(14,16)(15,20)(21,31)(22,28)(23,30)(24,32)(25,33)(26,34)(27,29)(35,38)(36,40)(37,39);(10,16)(13,14)(17,30)(18,33)(19,20)(22,23)(25,27)(28,32)(29,34)(31,38)(35,37)(39,40);(2,5)(6,8)(9,11,15,24)(10,17,20,32)(12,21,26,36)(13,23,14,22)(16,28,19,30)(18,31,34,40)(25,37,27,35)(29,38,33,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,25)(14,27)(15,26)(16,29)(17,31)(19,33)(20,34)(22,35)(23,37)(24,36)(28,38)(30,39)(32,40);(9,13,15,14)(10,19,20,16)(11,22,24,23)(12,25,26,27)(17,28,32,30)(18,33,34,29)(21,35,36,37)(31,38,40,39)

Polycyclic presentation:

2^9|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, 6355971);