#8096499

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

4328 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:

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

Polycyclic presentation:

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

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, 8003695);