#8087891

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	#49177	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,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, 7995087);