#8665288

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

4419 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,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, 8572484);