#8659372

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

4700 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8566568);