#8075289

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

4267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=g,f^2=g,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, 7982485);