#8096904

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

4296 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,e^2=gh,f^2=gh,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, 8004100);