#6448665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
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	#23402	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3288 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)(12,26)(13,18)(14,21)(15,20)(16,19)(22,39)(23,32)(24,37)(25,34)(27,35)(28,36)(29,40)(30,38)(31,33);(9,11)(10,17)(12,27)(13,29)(14,24)(15,25)(16,31)(18,35)(19,33)(20,34)(21,37)(22,28)(23,30)(26,40)(32,36)(38,39);(2,5)(6,8)(9,12,15,13)(10,18,20,26)(11,22,25,23)(14,28,16,30)(17,32,34,39)(19,36,21,38)(24,29,31,27)(33,40,37,35);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,18,20,26)(11,23,25,22)(14,30,16,28)(17,32,34,39)(19,36,21,38)(24,27,31,29)(33,40,37,35);(9,14,15,16)(10,19,20,21)(11,24,25,31)(12,28,13,30)(17,33,34,37)(18,36,26,38)(22,29,23,27)(32,40,39,35)

Polycyclic presentation:

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

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, 6355861);