#8678954

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

4404 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8586150);