#6449945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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)(9,10,16,22,15,11,14,19)(12,20,27,23,25,17,26,33)(13,18,31,34,30,21,29,28)(24,35,39,36,37,32,38,40);(2,5)(6,8)(9,11,15,10)(12,23,25,33)(13,28,30,34)(14,19,16,22)(17,27,20,26)(18,31,21,29)(24,35,37,32)(36,39,40,38);(2,5)(6,8)(9,12)(10,17)(11,20)(13,24)(14,26)(15,25)(16,27)(18,32)(19,23)(21,35)(22,33)(28,36)(29,38)(30,37)(31,39)(34,40);(1,3,4,7)(2,6,5,8)(9,13,15,30)(10,18,11,21)(12,24,25,37)(14,29,16,31)(17,32,20,35)(19,28,22,34)(23,36,33,40)(26,38,27,39);(12,25)(17,20)(23,33)(24,37)(26,27)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6357141);