#8687344

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

4303 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)(3,5)(4,7)(6,8)(9,10,16,19)(11,20,15,17)(12,14,13,18)(21,23,22,24)(25,26)(27,32)(28,34)(29,33)(30,35)(31,36)(37,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11,16,15)(10,17,19,20)(12,22,13,21)(14,23,18,24)(25,27,31,29)(26,32,36,33)(28,39,35,37)(30,38,34,40);(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,23,20,24)(25,27,31,29)(26,33,36,32)(28,37,35,39)(30,38,34,40);(1,4,6,5)(2,7,8,3)(9,13,16,12)(10,14,19,18)(11,22,15,21)(17,24,20,23)(25,28,31,35)(26,34,36,30)(27,37,29,39)(32,38,33,40);(1,5,6,4)(2,3,8,7)(9,14)(10,12)(11,23)(13,19)(15,24)(16,18)(17,22)(20,21)(25,29,31,27)(26,32,36,33)(28,39,35,37)(30,40,34,38);(1,3)(2,5)(4,8)(6,7)(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,24,18,23)(25,30,31,34)(26,35,36,28)(27,38,29,40)(32,37,33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=h,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, 8594540);