#8334304

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

4508 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8241500);