#8057868

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	#40378	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4265 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)(11,14)(12,15)(13,18)(16,22)(17,20)(19,24)(21,23)(27,29)(31,33)(35,36)(38,39);(1,2)(3,7)(4,8)(5,6)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,26,30,34)(27,31,29,33)(28,32,37,40)(35,38,36,39);(1,3,5,4)(2,7,6,8)(10,11)(15,17)(16,18)(23,24)(25,27,30,29)(26,31,34,33)(28,36,37,35)(32,39,40,38);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,28,30,37)(26,32,34,40)(27,35,29,36)(31,38,33,39);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29,30,27)(26,33,34,31)(28,36,37,35)(32,39,40,38)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965064);