#6491109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470237
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3289 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398305);