#7979810

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
prime radical	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4267 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)(11,17)(12,16)(13,15)(14,19)(18,23)(20,22)(21,24)(25,26,29,30)(27,33,34,31)(28,32,37,36)(35,39,40,38);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(10,15)(14,17)(16,22)(21,23)(26,30)(27,34)(32,36)(35,40);(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,22)(23,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7887006);