#7714473

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

4501 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)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,31,34)(27,35,36,32)(28,30,39,40)(29,33,37,38);(1,3,5,8)(2,6,4,7)(9,10,13,14)(11,15,18,17)(12,16,21,20)(19,22,24,23)(25,27)(26,32)(28,29)(30,38)(31,36)(33,40)(34,35)(37,39);(10,14)(11,18)(16,20)(19,24)(27,36)(29,37)(32,35)(33,38);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28)(26,30)(27,37)(29,36)(31,39)(32,33)(34,40)(35,38);(1,4,5,2)(3,7,8,6)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29,31,37)(26,33,34,38)(27,28,36,39)(30,32,40,35);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,28)(27,38)(29,32)(31,40)(33,36)(34,39)(35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621669);