#8688333

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

4293 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)(3,7,8,5)(12,20)(15,23)(19,24)(21,22)(25,26,31,30)(27,35,37,32)(28,33,36,38)(29,34,39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,21)(20,23)(22,24)(25,27,31,37)(26,32,30,35)(28,29,36,39)(33,40,38,34);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,19)(14,17)(15,22)(16,18)(20,24)(21,23)(25,28,31,36)(26,33,30,38)(27,29,37,39)(32,40,35,34);(1,3)(2,5)(4,8)(6,7)(9,12,14,20)(10,15,18,23)(11,19,17,24)(13,22,16,21)(25,29)(26,34)(27,36)(28,37)(30,40)(31,39)(32,33)(35,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,18)(14,17)(15,21)(20,24)(22,23)(25,29)(26,34)(27,36)(28,37)(30,40)(31,39)(32,33)(35,38);(9,13,14,16)(10,17,18,11)(12,21,20,22)(15,19,23,24)(25,30,31,26)(27,35,37,32)(28,38,36,33)(29,34,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=hi,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, 8595529);