#7940417

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

4454 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)(9,10,15,17)(11,19,20,16)(12,22,23,14)(13,24,21,18)(25,26,30,33)(27,34,31,28)(29,32,37,39)(35,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,11,15,20)(10,16,17,19)(12,21,23,13)(14,18,22,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,38)(39,40);(10,17)(11,20)(12,23)(18,24)(25,28,30,34)(26,31,33,27)(29,36,37,40)(32,38,39,35);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,21)(13,20)(15,23)(16,18)(17,22)(19,24)(25,28)(26,27)(29,36)(30,34)(31,33)(32,35)(37,40)(38,39);(9,13,15,21)(10,18,17,24)(11,12,20,23)(14,16,22,19)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,38)(39,40);(9,14)(10,12)(11,18)(13,19)(15,22)(16,21)(17,23)(20,24)(25,29,30,37)(26,32,33,39)(27,35,31,38)(28,36,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7847613);