#6447729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3022 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,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, 6354925);