#7973932

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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)(9,10,14,16)(11,17,12,15)(13,18,21,23)(19,24,20,22)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,40,37);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,15,16,17)(13,20,21,19)(18,24,23,22)(25,27)(26,31)(28,37)(29,39)(30,36)(32,40)(33,38)(34,35);(1,3,5,4)(2,7,6,8)(10,16)(11,12)(18,23)(19,20)(27,36)(28,38)(31,35)(32,39);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,28,30,38)(26,32,34,39)(27,37,36,33)(29,31,40,35);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,35,38,31)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7881128);