#8037860

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

4588 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)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(26,30)(28,36)(31,33)(34,37);(1,2)(3,6)(4,5)(7,8)(11,17)(14,21)(18,23)(22,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26,29,30)(27,31,35,33)(28,32,36,38)(34,39,37,40);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27,29,35)(26,31,30,33)(28,37,36,34)(32,40,38,39);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,28,29,36)(26,32,30,38)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=g,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, 7945056);