#6459603

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2268 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)(3,5)(4,6)(7,8)(9,10,14,17,15,19,26,18)(11,20,23,31,24,30,22,13)(12,25,28,16,29,32,39,33)(21,35,37,40,38,27,36,34);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,21)(14,22)(15,24)(16,27)(17,30)(18,20)(19,31)(23,26)(25,34)(28,36)(29,38)(32,40)(33,35)(37,39);(2,6)(3,7)(9,12,26,39,15,29,14,28)(10,16,17,32,19,33,18,25)(11,21,23,37,24,38,22,36)(13,27,30,40,31,35,20,34);(3,7)(5,8)(9,13)(10,11)(12,27)(14,20)(15,31)(16,21)(17,23)(18,22)(19,24)(25,36)(26,30)(28,34)(29,35)(32,37)(33,38)(39,40);(9,14,15,26)(10,17,19,18)(11,22,24,23)(12,28,29,39)(13,30,31,20)(16,32,33,25)(21,36,38,37)(27,40,35,34)

Polycyclic presentation:

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

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, 6366799);