#8075533

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

4487 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7982729);