#7988204

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

4398 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)(9,10,15,17)(11,16,12,14)(13,21,22,18)(19,24,20,23)(27,34)(32,38)(35,40)(36,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,23,21,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,39)(36,40);(1,2,5,4)(3,6,8,7)(10,17)(11,12)(13,22)(23,24)(26,31)(28,37)(32,38)(35,40);(9,12,15,11)(10,16,17,14)(13,20,22,19)(18,23,21,24)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,24)(15,22)(16,23)(17,21)(25,28,30,37)(26,33,31,29)(27,35,34,40)(32,39,38,36);(9,14,15,16)(10,11,17,12)(13,23,22,24)(18,20,21,19)(25,29,30,33)(26,28,31,37)(27,36,34,39)(32,35,38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=i,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, 7895400);