#9515359

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
prime radical	#51865	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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:

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

Polycyclic presentation:

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

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, 9422555);