#7877879

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

4518 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7785075);