#9788883

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

4308 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9696079);