#8057935

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

4177 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)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,26)(27,29)(28,31)(30,36)(32,34)(33,39)(35,38)(37,40);(1,3,5,8)(2,6,4,7)(10,15)(12,16)(18,21)(20,23)(25,27)(26,32)(28,29)(30,35)(31,34)(33,38)(36,37)(39,40);(1,2,5,4)(3,6,8,7)(10,15)(11,17)(18,21)(19,24)(26,28)(27,34)(33,36)(35,40);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,28,31,26)(27,29,34,32)(30,36,39,33)(35,37,40,38);(1,2,5,4)(3,6,8,7)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29)(26,27)(28,34)(30,38)(31,32)(33,40)(35,36)(37,39);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,30,31,39)(26,33,28,36)(27,35,34,40)(29,37,32,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,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, 7965131);