#7891213

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

4440 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)(11,12)(13,17)(14,18)(15,16)(19,20)(21,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,15,18,12)(13,19,22,23)(17,21,24,20)(25,26,31,30)(27,35,38,32)(28,33,37,39)(29,34,36,40);(9,10,14,18)(11,12,16,15)(13,17,22,24)(19,20,23,21)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,34)(39,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,23)(22,24)(25,28,31,37)(26,33,30,39)(27,36,38,29)(32,34,35,40);(9,12,14,15)(10,16,18,11)(13,21,22,20)(17,19,24,23)(25,29,31,36)(26,34,30,40)(27,37,38,28)(32,33,35,39);(1,3,4,7)(2,6,5,8)(9,13,14,22)(10,17,18,24)(11,19,16,23)(12,20,15,21)(25,30,31,26)(27,35,38,32)(28,39,37,33)(29,34,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7798409);