#8093297

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

4536 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000493);