#7987876

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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7895072);