#8087814

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
prime radical	#44417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4255 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7995010);