#8688325

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4331 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595521);