#9720206

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

4535 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)(3,6)(4,7)(5,8)(9,10,14,15)(11,16,19,23)(12,17,22,21)(13,20,24,18)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(1,3)(2,5)(4,8)(6,7)(10,15)(12,22)(16,23)(18,20)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(10,15)(11,19)(17,21)(18,20)(25,27,29,35)(26,30,32,33)(28,34,37,36)(31,38,40,39);(9,11)(10,16)(12,20)(13,21)(14,19)(15,23)(17,24)(18,22)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,20)(13,23)(14,22)(15,21)(16,24)(18,19)(25,28)(26,31)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,2)(3,5)(4,7)(6,8)(9,13,14,24)(10,18,15,20)(11,21,19,17)(12,16,22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9627402);