#9515351

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	#47231	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

(10,15)(11,18)(13,21)(20,23)(25,26,31,36)(27,32,29,34)(28,38,39,33)(30,35,37,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,12,18,16)(13,17,21,22)(19,20,24,23)(25,27,31,29)(26,32,36,34)(28,30,39,37)(33,40,38,35);(2,5)(3,7)(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,28,31,39)(26,33,36,38)(27,37,29,30)(32,35,34,40);(3,7)(6,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(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)(10,11)(13,20)(14,16)(15,18)(17,19)(21,23)(22,24)(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)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(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=h,d^2=gh,e^2=gi,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, 9422547);