#8687365

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

4299 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 8594561);