#8839180

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

4366 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746376);