#8057931

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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)(9,10,16,20)(11,17,22,23)(12,13,14,18)(15,21,24,19)(25,26,31,34)(27,35,39,32)(28,29,38,40)(30,33,36,37);(1,3,5,8)(2,6,4,7)(9,11,16,22)(10,17,20,23)(12,19,14,21)(13,15,18,24)(25,27,31,39)(26,32,34,35)(28,30,38,36)(29,37,40,33);(9,12,16,14)(10,18,20,13)(11,21,22,19)(15,23,24,17)(25,28)(26,29)(27,36)(30,39)(31,38)(32,33)(34,40)(35,37);(9,13,16,18)(10,12,20,14)(11,15,22,24)(17,19,23,21)(25,29)(26,28)(27,37)(30,32)(31,40)(33,39)(34,38)(35,36);(1,2,5,4)(3,6,8,7)(9,14,16,12)(10,18,20,13)(11,19,22,21)(15,23,24,17)(25,28)(26,29)(27,30)(31,38)(32,37)(33,35)(34,40)(36,39);(1,4,5,2)(3,7,8,6)(9,15)(10,19)(11,18)(12,23)(13,22)(14,17)(16,24)(20,21)(25,30)(26,33)(27,38)(28,39)(29,32)(31,36)(34,37)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965127);