#8974559

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8881755);