#8688337

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

4453 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=ghi,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, 8595533);