#6449189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3402 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,26)(13,18)(14,31)(15,20)(16,19)(17,25)(22,39)(23,36)(24,35)(27,33)(28,37)(29,40)(30,38)(32,34);(9,11,14,25,15,24,16,32)(10,17,19,21,20,34,31,35)(12,27,28,22,13,29,30,23)(18,36,37,40,26,39,38,33);(2,5)(6,8)(9,12,15,13)(10,18,20,26)(11,22,24,23)(14,28,16,30)(17,33,34,40)(19,37,31,38)(21,36,35,39)(25,29,32,27);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,18,20,26)(11,23,24,22)(14,30,16,28)(17,33,34,40)(19,37,31,38)(21,36,35,39)(25,27,32,29);(11,24)(17,34)(21,35)(22,23)(25,32)(27,29)(33,40)(36,39)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6356385);