#8678889

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

4459 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,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, 8586085);