#8081032

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

4592 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)(12,17)(13,23)(14,16)(18,20)(19,24)(21,22)(25,26,31,35)(27,32,36,40)(28,30,34,39)(29,38,37,33);(1,3,5,8)(2,6,4,7)(9,11,14,19)(10,15,16,24)(12,20,22,13)(17,18,21,23)(25,27)(26,32)(28,37)(29,34)(30,33)(31,36)(35,40)(38,39);(1,2,5,4)(3,6,8,7)(10,16)(11,19)(13,20)(17,21)(27,36)(29,37)(32,40)(33,38);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,13)(14,22)(15,23)(16,21)(18,24)(19,20)(25,28)(26,30)(27,29)(31,34)(32,38)(33,40)(35,39)(36,37);(1,2,5,4)(3,6,8,7)(9,13)(10,18)(11,12)(14,20)(15,21)(16,23)(17,24)(19,22)(25,29)(26,33)(27,34)(28,36)(30,32)(31,37)(35,38)(39,40);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,21)(13,18)(14,16)(17,22)(19,24)(20,23)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,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, 7988228);