#7987595

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	#34715	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4469 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=gi,e^2=h,f^2=i,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, 7894791);