#6459600

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

2332 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)(3,5)(4,6)(7,8)(9,10,13,17,14,19,21,18)(11,20,23,15,24,28,34,29)(12,16,26,31,25,30,35,32)(22,33,37,27,36,39,40,38);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,19)(17,21)(20,23)(22,27)(24,29)(25,30)(26,32)(28,34)(31,35)(33,37)(36,38)(39,40);(2,6)(3,7)(9,11,21,34,14,24,13,23)(10,15,17,28,19,29,18,20)(12,22,35,40,25,36,26,37)(16,27,31,39,30,38,32,33);(3,7)(5,8)(9,12)(10,16)(11,22)(13,26)(14,25)(15,27)(17,31)(18,32)(19,30)(20,33)(21,35)(23,37)(24,36)(28,39)(29,38)(34,40);(12,25)(16,30)(22,36)(26,35)(27,38)(31,32)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,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, 6366796);