#7864992

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=gh,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, 7772188);