#9203761

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

4543 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)(3,8)(4,7)(5,6)(9,10,14,12)(11,15,18,16)(13,17,21,20)(19,22,24,23)(25,26,31,34)(27,28,29,32)(30,38,39,33)(35,40,37,36);(1,2,6,5)(3,7,4,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,27)(26,28)(29,31)(30,35)(32,34)(33,36)(37,39)(38,40);(1,3,6,4)(2,7,5,8)(11,18)(15,16)(19,24)(22,23)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(3,4)(7,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,38)(39,40);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,16,18,15)(13,17,21,20)(19,22,24,23)(25,29,31,27)(26,32,34,28)(30,35,39,37)(33,36,38,40);(1,5,6,2)(3,8,4,7)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110957);