#8080798

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

4450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=gh,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, 7987994);