#9016580

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

4635 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)(4,6)(5,8)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(1,2,4,6)(3,5,8,7)(10,12)(15,17)(16,20)(22,23)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(10,12)(11,18)(13,21)(22,23)(25,28,30,34)(26,31,33,27)(29,37,38,36)(32,35,40,39);(1,3,4,8)(2,5,6,7)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,28)(26,27)(29,37)(30,34)(31,33)(32,39)(35,40)(36,38);(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,29,30,38)(26,32,33,40)(27,35,31,39)(28,36,34,37);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=hi,e^2=gi,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, 8923776);