#7901319

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=i,e^2=ghi,f^2=i,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, 7808515);