#8689959

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

4606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8597155);