#7987972

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	#36141	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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)(9,10,15,14)(11,19,20,16)(12,22,21,17)(13,18,23,24)(26,30)(27,33)(32,36)(35,39);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,14,19)(12,13,21,23)(17,18,22,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(11,20)(13,23)(16,19)(18,24)(25,26,29,30)(27,34,33,31)(28,32,37,36)(35,40,39,38);(1,2,5,4)(3,6,8,7)(9,12,15,21)(10,17,14,22)(11,13,20,23)(16,18,19,24)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,21)(12,20)(14,24)(15,23)(16,22)(17,19)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(9,14,15,10)(11,16,20,19)(12,22,21,17)(13,18,23,24)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=gi,f^2=i,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, 7895168);