#8081437

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

4323 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7988633);