#7927539

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

4398 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834735);