#6583829

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574086
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3045	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491025);