#8037834

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

4374 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

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, 7945030);