#8075455

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

4394 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)(11,12)(13,17)(14,15)(16,19)(18,20)(21,23)(22,24)(25,26,31,29)(27,34,28,32)(30,33,38,37)(35,40,36,39);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,12,16,19)(13,17,21,23)(18,20,22,24);(10,15)(12,19)(13,21)(18,22)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(9,11,14,16)(10,12,15,19)(13,22,21,18)(17,24,23,20)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,15)(13,20)(14,19)(17,22)(18,23)(21,24)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,17,15,23)(11,18,16,22)(12,20,19,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7982651);