#10045609

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=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, 9952805);