#7868925

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7776121);